OF   THE 


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af  California, 


Book  and  Volume, 


i87</~ 


HYDRAULIC    MOTORS 


TRANSLATED   FROM  TUB 


FRENCH  COURS  DE  MECANIQUE  APPLIQUEE. 


PAR 

M.    BRESSE 

Professeur  de  Mfaanique  a  V^cole  des  Fonts  et  Cfiausstes. 


F.  A.  MAHA1ST, 

LIEUTENANT  TT.   8.   CORPS   OF  ENGINEERS. 


REVISED  BY 


D.  H.  MAHAN,  LL.D., 

PROFESSOR  OF  CIVIL  ENGINEERING,   AC.,   UNITED   STATES   MILITARY   ACADEMY. 


NEW  YORK : 

JOHN  WILEY  &  SON,  2  CLINTON  HALL,  ASTOR  PLACE. 

1869. 


Entered  according  to  Act  of  Congress,  in  the  year  1869,  by 

F.  A.  MAHAN, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Southern  District  of 

New  York. 


THE  NEW  YORK  PRINTING  COMPANY, 

8 1,  83,  and  85  Centre  Street, 

NEW  YORK. 


PREFACE. 


THE  eminent  position  of  M.  Bresse  in  the  scientific 
world,  and  in  the  French  Corps  of  Civil  Engineers,  is 
my  best  apology  for  attempting  to  supply  a  want,  felt 
by  the  students  of  civil  engineering  in  our  country,  of 
some  standard  work  on  Hydraulic  Motors,  by  furnish- 
ing a  translation  of  the  chapter  on  this  subject  con- 
tained in  the  second  volume  of  M.  Bresse's  lectures 
on  Applied  Mechanics,  delivered  to  the  pupils  of  the 
School  of  Civil  Engineers  (I&cole  des  Ponts  et  CJiaus- 
sees)  at  Paris. 

In  making  the  translation,  I  have  retained  the  French 
units  of  weights  and  measures  in  the  numerical  exam- 
ples given,  as  the  majority  of  our  students  are  conver- 
sant with  them. 

F.  A.  MAHAK 

WILLETT'S  POINT,  K  Y.,  July,  1869. 


CONTENTS. 


§  I.— PRELIMINARY  IDEAS  ON  HYDRAULIC  MOTORS. 

ART.    PAGK. 

Definitions;  theorem  of  the  transmission  of  work  in  machines 1  9 

Analogous  ideas  applied  to  a  waterfall 2  11 

General  observations  on  the  means   of  securing  a  good  effective 

delivery  for  a  waterfall  driving  a  hydraulic  motor 3  14 


§  II.— WATER  WHEELS  WITH  A  HORIZONTAL  AXLE. 

Undershot  wheel  with  plane  buckets  or  floats  moving  in  a  confined 

race 4  17 

Wheels  arranged  according  to  Poncelet's  method 5  28 

Paddle  wheels  in  an  unconfined  current 6  34 


BREAST  WHEELS. 

Wheels  set  in  a  circular  race,  called  breast  wheels 7  36 

Example  of  calculations  for  a  rapidly  moving  breast  wheel 8  48 

OVERSHOT  WHEELS. 

Wheels  with  buckets,  or  overshot  wheels 9  51 

§  III.— WATER  WHEELS  WITH  VERTICAL  AXLES. 

Old-fashioned  spoon  or  tub  wheels 10  66 

TURBINES. 

Of  turbines : 11  69 

Fourneyron's  turbine 12  70 

Fontaine's  turbine 13  73 

Koscklin's  turbine 14  76 

Theory  of  the  three  preceding  turbines 15  78 

Remarks  on  the  angles  /?,  y,  0,  and  on  the  dimensions  5,  £',  r,  r', 

h.  h'...  .  16  90 


yiii  CONTENTS. 

ART.  PAGE. 

Examples  of  the  calculations  to  be  made  in  constructing  a  turbine. .  17  93 

Of  the  means  of  regulating  the  expenditure  of  water  in  turbines.  ..  18  98 

Hydropneumatic  turbine  of  Girard  and  Gallon 19  103 

Some  practical  views  on  the  subject  of  turbines 20  105 

Reaction  wheels. .                                                                              ....  21  107 


§  IV.— OF  A  FEW  MACHINES  FOR  RAISING  WATER. 

Pumps 22  111 

Spiral  noria 23  122 

Lifting  turbines  ;  centrifugal  pump 24  127 

Authorities  on  Water  Wheels. .  134 


APPENDIX. 

Comparative  table  of  French  and  United  States  measures 135 

Note  A,  Article  1 135 

Note  B,  Article  2 136 

Note  C,  Article  4 137 

Note  D,  Article  9 140 

Note  E,  Boyden  turbines,  from  Lowell  Hydraulic  Experiments,  by 

James  B.  Francis,  Esq 141 


HYDRAULIC  MOTORS, 


AND   SOME 


MACHINES    FOR  RAISING  WATER, 


§  |. — PRELIMINARY  IDEAS  ON  HYDRAULIC  MOTORS. 

1.  Definitions  y  theorem  of  the  transmission  of  work  in  ma- 
chines.— The  term  machine  is  applied  to  any  body  or  collection 
of  bodies  intended  to  receive  at  some  of  their  points  certain 
forces,  and  to  exert,  at  other  points,  forces  which  generally 
differ  from  the  first  in  their  intensity,  direction,  and  the  velocity 
of  their  points  of  application. 

The  dynamic  effect  of  a  machine  is  the  total  work,  generally 
negative  or  resisting,  which  it  receives  from  external  bodies 
subject  to  its  action.  It  happens  that  the  dynamic  effect  is 
sometimes  positive  work:  for  example,  when  we  let  down  a 
load  by  a  rope  passed  over  a  pulley,  the  weight  of  the  load  pro- 
duces a  motive  work  on  the  system  of  the  rope  and  pulley. 

Let  us  suppose,  to  make  this  clear,  that  the  dynamic  effect  is 
a  resisting  work.  Independently  of  this,  the  machine  is  affected 
by  some  others  which  are  employed  to  overcome  friction,  the 
resistance  of  the  air,  &c.  The  resistances  which  give  rise  to 
these  negative  works  have  received  the  name  of  secondary 


10  GENERAL    THEOREM. 

resistances;  whilst  the  dynamic  effect  is  due  to  what  are  called 
principal  resistances. 

The  general  theorem  of  mechanics,  in  virtue  of  which  a 
relation  is  established  between  the  increase  of  living  force  of 
a  material  system  and  the  work  of  the  forces,  is  applicable 
to  a  machine  as  to  every  assemblage  of%  bodies.  To  express 
it  analytically,  let  us  suppose  the  dynamic  effect  to  be  taken 
negative,  and  let  us  call 

Tw  the  sum  of  the  work  of  the  motive  forces  which  have 
acted  on  the  machine  during  a  certain  interval  of  time  ; 
T0  the  dynamic  effect  during  the  same  time  ; 
Ty-,  the  corresponding  value  of  the  work  of  the  secondary 

resistances  ; 

v  and  v0,  the  velocities  of  a  material  point,  whose  mass  is  m, 
making  a  part  of  the  machine,  at  the  beginning  and  end 
of  the  time  in  question  ; 
H  and  H0,  the  corresponding  distances  of  the  centre  of  grav- 

ity of  the  mechanism  below  a  horizontal  plane  ; 
2,  a  sum  extended  to  all  the  masses  m. 
From  the  theorem  above  mentioned  there  obtains 


In  a  machine  moving  regularly,  each  of  the  velocities  v  in- 
creases from  zero,  the  value  corresponding  to  a  state  of  rest,  to 
a  certain  maximum  which  it  never  exceeds  ;  the  first  member 
of  the  equation  has,  then,  necessarily,  a  superior  limit,  below 
which  it  will  be  found,  or  which  it  will,  at  the  farthest,  reach, 
whatever  be  the  interval  of  time  to  which  the  initial  and  final 
values  v0  and  v  are  referred.  The  same  holds  •with  the  term 
(H  —  H0)  2  m  g,  when  the  machine  moves  without  changing 
its  place,  as  its  centre  periodically  occupies  the  same  positions. 
On  the  contrary,  the  terms  Tm,  Te,  Tf  increase  indefinitely  with 
the  time,  if  the  motion  of  the  machine  is  prolonged,  because 


GENERAL    THEOREM.  11 

new  quantities  of  work  are  being  continually  added  to  those 
already  accumulated.  These  terms  will  at  length  greatly 
surpass  the  others,  so  the  equation,  therefore,  should,  after  an 
unlimited  interval  of  time,  reduce  to 

Tm  =  Te  +  T,, 

This  is  what  would  really  take  place,  without  supposing  the 
time  unlimited,  if  the  beginning  and  end  of  this  interval  cor- 
responded to  a  state  of  rest  of  the  machine,  and  if,  at  the  same 
time,  H  =  H0.  We  can  then  say  that,  as  a  general  rule,  the 
motive  work  is  equal  to  the  resisting  work ;  but  as  this  last 
includes,  besides  the  dynamic  effect  for  which  the  machine  is 
established,  the  work  of  the  secondary  resistances,  we  see  that 
the  action  of  the  motive  power  is  not  all  usefully  employed, 
since  a  portion  goes  to  overcoming  the  work  of  Tf. 

T 

It  is  evident  that  the  ratio  ?£-   measures    the    proportional 

*f» 

T  T 

loss;  the  ratio  7^,  or  1  —  7=^-,  gives,  on  the  contrary,  a  clear 

-*-m  -*-m 

idea  of  the  portion  of  the  effective  work.  This  last  ratio  is 
what  is  called  the  delivery  of  the  machine;  it  is  evidently 
always  less  than  unity,  since  in  the  best-arranged  machines  Tf 
still  preserves  a  certain  value.  The  skill  of  the  constructor  is 
shown  in  bringing  this  as  near  unity  as  possible. 

2.  Analogous  ideas  applied  to  a  waterfall. — A  waterfall 
may  be  considered  as  a  current  of  water  flowing  through  two 
sections  C  B,  E  F  (Fig.  1),  at  no  great  distance  apart,  but 
with  a  noticeable  difference  of  level,  H,  between  the  surface 
slope  at  C  and  E,  and  which  may  be  assumed  to  yield  a 
constant  volume  of  water  for  each  unit  of  time.  The  mate- 
rial liquid  system  thus  comprised  between  the  sections  C  B, 
E  F,  at  each  instant  may  be  regarded  as  a  machine  that  is  con- 
tinually renewed,  the  molecules  which  flow  out  through  E  F' 


12 


GENEEAL    THEOEEM. 


being  replaced  by  those  which  enter  through  C  B.  The  motive 
work  in  this  case  will  be  that  of  the  weight  combined  with  that 
of  the  pressures  on  the  external  boundaries  of  the  system ;  the 
dynamic  effect  will  be  the  work  of  the  resistances  against  the 
fall  of  water  caused  by  any  apparatus  whatever,  a  water-wheel 


FIG.  1. 

for  example,  exposed  to  its  action.  In  its  turn  the  water-wheel, 
considered  as  a  machine,  will  receive  from  the  fall  a  motive 
work  sensibly  equal  to  the  dynamic  effect  that  we  have  just 
spoken  of,*  and  will  change  only  a  portion  of  it  into  useful 
work,  which  will  be  its  dynamic  effect  proper.  But  in  what  is 
to  follow  we  shall  limit  ourselves  to  studying  the  dynamic  effect 
of  the  fall,  and  not  that  of  the  wheel. 

Although  the  motion  of  the  liquid  cannot  always  be  strictly 
the  same,  because  the  wheel  does  not  always  maintain  exactly 
the  same  position,  still  it  can  be  so  regarded  without  material 
error;  for  after  an  interval  of  time  d,  generally  very  short, 
occupied  by  a  float  or  paddle  in  taking  the  place  of  the  one  that 
preceded  it,  everything  returns  to  the  same  condition  as  at  the 
beginning  of  the  interval.  Supposing  the  motion  of  the  wheel 
regular  and  the  paddles  to  be  uniformly  distributed,  there  is 
such  a  frequent  periodicity  in  the  state  of  the  system  that  it 

*  We  say  sensibly,  for  the  equality  between  the  mutual  actions  of  the  watei 
and  wheel  do  not  involve  that  of  corresponding  work.  This  equality  is  only 
strict  in  supposing  the  friction  of  the  liquid  against  the  solid  sides  of  the 
wheel  zero,  which  friction  is  in  reality  very  slight. 


GENERAL   THEOKEM.  13 

almost  amounts  to  a  permanency.  We  will  now  apply  Ber- 
nouilli's  theorem  to  any«nolecule  whatever,  having  a  mass  m, 
which,  departing  from  the  section  C  B  with  the  velocity  Y0, 
reaches  E  F  with  a  velocity  Y.  The  entire  head  is  H,  if  we 
allow  the  parallelism  of  the  threads  in  the  extreme  sections,  for 
the  pressure  then  varies  according  to  the  hydrostatic  law  in  each 
of  the  two  surfaces  C  B  and  E  F,  so  that  the  points  C  and  E 
can  be  considered  as  piezometric  levels  for  the  initial  and  final 
positions  of  the  mass  m.  Let  —te  and  —  ^be  the  respective 
work  referred  to  the  unit  of  mass,  and  considered  as  resistances 
which  m  has  encountered  in  its  course  between  C  B  and  E  F, 
in  consequence  of  the  action  of  the  wheel  and  of  viscosity. 
Then  from  the  general  theorem  of  living  forces  we  have  the 
equation  — 

i  Y2  —  Y«2 

*-,-ft+fr)-  -^--  =  °' 

whence, 

"\r  a        Y2  \ 

m  te  =  m  g  (H   +   ~  -  ^)  m  tf. 

Consequently  we  see  that  the  weight  m  g  of  each  molecule 
which  passes  from  C  B  to  E  F  gives  rise  to  a  dynamic  effect 
m  te,  the  value  of  which  is  expressed  in  the  second  member  of 
the  equation.  The  sum  of  the  weights  of  the  molecules  m  in- 
cluded in  the  entire  weight  P,  which  the  current  expends  in  a 
second,  will  produce  a  dynamic  effect  equal  to  a  sum  2  of  analo- 
gous expressions  extended  to  all  these  masses  ;  considering  Y0 
and  Y  as  constant  velocities  in  the  respective  sections  0  B  and 
E  F,  this  summation  will  give  — 


Moreover,  in  each  second  a  new  weight  P  is  supplied  by  the 
current  ;  there  is  then  produced  a  new  dynamic  effect  2  m  t«, 
which  thus  represents  the  mean  dynamic  effect  in  each  second. 


14  GENERAL   THEOREM. 

y a         y 2  v 

The  quantity  P  (  H  -f  ^ —  —  ^ — J  reduces  to  P  H,  in  the 

case  in  which  Y0  and  V  are  sensibly  equal  to  zero,  which  hap- 
pens in  measuring  the  difference  of  level  between  the  basins 
from  which  the  water  starts  and  that  into  which  it  flows,  when 
the  water  is  nearly  at  a  stand-still ;  we  then  call  the  product 

P  H  the  effective  delivery  of  the  fall.     The  ratio  - p      *   is  the 

productive  force ;  2  m  tf  is  the  work  lost.  Dividing  the  last 
equation  by  P  or  2  m  g,  and  supposing  Y  =  0,  Y0  =  0,  there 
obtains, 

g  2  m  g  2  m 

H  in  this  expression  may  be  regarded  as  the  total  head  of  water ; 

— - — e  the  productive  head,  that  is,  the  height  which,  multi- 
plied by  the  weight  P  expended,  would  give  the  dynamic  effect 

per  second ;   -       '—  the  head  lost,  to  be  subtracted  from  the 
g  2,  m 

total  head  when  the  productive  head  is  required.     We  see  that 

— - — f  is  the  mean  loss  of  head  experienced  by  the  molecules  in 
y 

their  passage  from  C  B  to  E  F ;  for  this  expression  represents 
exactly  the  mean  work  of  viscosity  on  a  molecule,  referred  to 
the  unit  of  mass  and  divided  by  g. 

3.  General  remarks  on  the  means  of  securing  a  satisfactory 
delivery  from  a  head  of  water  which  moves  an  hydraulie 
motor. — In  order  to  obtain  a  good  satisfactory  delivery,  we 
must  seek  to  diminish  as  much  as  possible  the  term  2  m  tf,  or 

the  mean  loss  of  head —f-     A  few  of  the  causes  that  pro- 

g  2  m 

duce  this  loss  will  now  be  pointed  out,  and  the  manner  in 
which  they  may  be  diminished. 


GENERAL    THEOREM.  15 

Firstly,  if  the  water  enters  the  wheel,  and  can  in  consequence 
act  on  it,  it  is  because  it  possesses  a  certain  relative  velocity  w  ; 
now,  in  the  majority  of  cases  this  relative  velocity  gives  rise  to 
a  violent  agitation  of  the  liquid  and  to  vibratory  motions,  from 

which  a  loss  of  head  is  experienced  equal  to  ^-,   like    to    what 

has  been  observed  to  obtain  in  the  collision  of  solid  bodies ; 

w* 
hence  ~-  would  be  a  portion  of  the  head  lost.*     It  is,  then, 

t/ 

generally  a  matter  of  importance  to  make  the  water  enter  with 
as  small  a  relative  velocity  as  possible.  However,  that  is  not 
necessary  when  w  has  its  direction  tangent  to  the  sides  that  it 
comes  in  contact  with,  and  when  the  particular  arrangement 
of  the  apparatus  allows  the  water  to  continue  its  relative  mo- 
tion in  the  wheel  without  there  being  any  shock  of  the  threads 
on  the  solid  sides  or  on  the  liquid  already  introduced,  since 
then  we  have  no  longer  to  fear  the  violent  disturbance  that  we 
have  just  spoken  of. 

When  the  water  on  leaving  the  wheel  is  received  into  a  race 
of  invariable  level,  in  which  it  loses  its  absolute  velocity  of 

*  Let  us  suppose  that  the  water  that  has  once  entered  the  wheel  passes  at 
once  to  relative  rest :  the  destruction  of  the  velocity  w  being  then  attributa- 
ble only  to  the  resisting  work  of  the  molecular  actions,  this  work  for  a  fluid 
molecule  having  the  mass  ra  would  be  |  m  w2,  a  quantity  which,  when  referred 

up 
to  the  unit  of  mass  and  divided  by  #,  would  give  the  loss  of  head  —  ,  which 

is  the  same  for  all  the  molecules.  But  the  supposition  of  the  instant  produc- 
tion of  relative  rest  is  not,  strictly  speaking,  true;  weight,  for  example,  can 
combine  sometimes  with  molecular  actions  to  bring  about  this  result  at  the 

«c2 
end  of  a  sensible  time.     Consequently  the  expression  <r-  must  be  considered 

*9 

less  as  the  exact  value  of  the  quantity  whose  value  we  here  wish  to  find,  than 
as  a  superior  limit  which  we  approach  more  or  less  nearly,  according  to  the 
particular  case  considered. 


16  GENERAL    THEOKEM. 

departure  v',  we  readily  see  that  this  velocity  v'  must  be  re- 
duced as  much  as  possible.  In  fact,  the  water  that  leaves  the 

P  vn 
wheel  carries  with  it,  in  each  second,  a  living  force,  -^ — > 

which  might  have  been  taken  up  by  a  resisting  work  of  the 
hydraulic  motor,  and  have  thus  increased  by  its  amount  the 
dynamic  effect  Te ;  whereas  this  living  force  will  only  serve  to 
produce  a  disturbance  and  eddies  in  the  interior  race,  and  will 
enter  the  term  2  m  tf.  There  are  cases,  however,  in  which  we 
are  obliged  to  give  v'  a  value  more  or  less  considerable ;  we 
shall  see  presently,  by  a  few  examples,  how  it  is  sometimes 
possible  to  diminish  this  unfavorable  condition. 

Ordinarily,  the  considerations  above  mentioned  are  under- 
stood when  we  say  that  the  water  must  enter  without  shock, 
and  leave  without  velocity.  We  can  also  add,  that  it  is  well 
not  to  deliver  it  too  rapidly  through  channels  of  too  little 
breadth,  as  this  would  involve  losses  of  head  to  be  included  in 

Xmtf 

the    expression  -  - — -. 
g  2  m 

We  shall  now  proceed  to  examine  the  most  widely  known 
hydraulic  motors,  keeping  principally  in  view  the  best  means 
of  making  use  of  the  head  in  each  caser  and  showing  the  man- 
ner of  calculating  the  dynamic  effect  that  can  be  realized  with 
the  means  adopted. 

Water-wheels  are  divided  into  two  great  classes — those  hav- 
ing a  vertical  and  those  having  a  horizontal  axis.  The  varieties 
included  in  these  two  classes  will  form  the  subject  of  the  fol- 
lowing paragraphs. 


v>c         '-*-»       '^J! 

Library. 

UNDERSHOT  WHEELS. 


§  1 1 .    WATEK- WHEELS  WITH  A  HORIZONTAL  Axis. 

4.  Undershot  wheel  with  plane  buckets  or  floats  moving  in  a 
confined  race. — These  wheels  are  ordinarily  constructed  of  wood. 
Upon  a  polygonal  arbor  A  (Fig.  2)  a  socket  0,  of  cast  iron, 


FIG. 


is  fastened  by  means  of  wooden  wedges  ft.  Arms  D  are  set 
in  grooves  cast  in  the  socket,  and  are  fastened  to  it  by  bolts ; 
these  arms  serve  to  support  a  ring  E  E,  the  segments  of  which 
are  fastened  to  each  other  and  to  the  arms  by  iron  bands.  In  the 
ring  are  set  the  projecting  pieces  FF ,  of  wood,  placed  at 


18  UNDERSHOT    WHEELS. 

equal  distances  apart,  and  intended  to  support  the  floats  G  G , 

which  are  boards  varying  from  Om.02  to  Om.03  in  thickness, 
situated  in  planes  passing  through  the  axis  of  the  wheel  and 
occupying  its  entire  breadth.  A  single  set  of  the  foregoing 
parts  would  not  be  sufficient  to  give  a  good  support  to  the 
floats.  In  wheels  of  little  breadth  in  the  direction  of  the  axis 
two  parallel  sets  will  suffice ;  if  the  wheels  are  broad,  three  or 
more  may  be  requisite. 

The  number  of  arms  increases  with  the  diameter  of  the 
wheel.  In  the  more  ordinary  kinds,  of  3  to  5  metres  in  dia- 
meter, each  socket  carries  six  arms.  The  floats  may  be  about 
Om.35  to  Om.40  apart,  and  have  a  little  greater  depth  in  the 
direction  of  the  radius,  say  Om.60  to  Om.70. 

From  this  brief  description  of  the  wheel,  let  us  now  see  how 
we  can  calculate  the  work  which  it  receives  from  the  head  of 
water.  The  water  flows  in  a  very  nearly  horizontal  current 
through  a  race  B  G  H  F  (Fig.  3),  of  nearly  the  same  breadth 


FFf 

FIG.  8. 


as  the  wheel,  a  portion  G  H  of  tlie  bottom  being  hollowed 
out,  in  a  direction  perpendicular  to  the  axis,  to  a  cylindrical 
shape,  and  allowing  but  a  slight  play  to  the  floats.  The  liquid 
molecules  have,  when  passing  C  B,  a  velocity  v,  but  shortly 
after  they  are  confined  in  the  intervals  limited  by  two  consecu- 
tive floats  and  the  race.  They  entered  these  spaces  with  a 
mean  relative  velocity  equal  to  the  difference  between  the  hori- 
zontal velocity  v,  and  the  velocity  v'  of  the  middle  of  the  im- 
mersed portion  of  the  floats,  the  direction  of  which  last  velocity 
is  also  very  nearly  horizontal.  There  result  from  this  relative 


UNDERSHOT    WHEELS.  19 

velocity  a  shock  and  disturbance  which  gradually  subside, 
while  the  floats  are  traversing  over  the  circular  portion  of  the 
canal ;  so  that  if  this  circular  portion  is  sufficiently  long,  and 
if  there  be  not  too  much  play  between  the  floats  and  the  canal, 
the  water  that  leaves  the  wheel  will  have  a  velocity  sensiblv 
equal  to  v'.  The  action  brought  to  bear  by  the  wheel  on  the 
water  is  the  cause  of  the  change  in  this  velocity  from  v  to  -y', 
which  gives  us  the  means,  as  we  shall  presently  see,  of  calculat- 
ing the  total  intensity  of  this  action. 

For  this  purpose  let  us  apply  to  the  liquid  system  included 
between  the  cross  sections  C  B,  E  F,  in  which  the  threads  are 
supposed  parallel,  the  theorem  of  quantities  of  motion  projected 
on  a  horizontal  axis.  Represent  by 

b  the  constant  breadth  of  the  wheel  and  canal ; 

A,  hr  the  depths  C  B,  E  F,  of  the  extreme  sections  which  are 
supposed  to  be  rectangular ; 

F  the  total  force  exerted  by  the  wheel  on  the  water,  or  in- 
versely, in  a  horizontal  direction  ; 

P  the  expenditure  of  the  current,  expressed  in  pounds,  per 
second ; 

n  the  weight  of  a  cubic  metre  of  the  water ; 

&  0  the  short  interval  of  time  during  which  C  B  E  F  passes 
to  C'  B'  E'  F'. 

The  liquid  system  C  B  E  F,  here  under  consideration,  is 
analogous  to  the  one  treated  in  Note  A  (see  Appendix),  in 
which  a  change  in  the  ^urface  level  takes  place ;  and  the  man- 
ner of  determining  the  gain  in  the  quantity  of  motion  during 
a  short  time  d,  and  calculating  the  corresponding  impulses, 
during  the  same  time,  produced  by  gravity  and  the  pressures 
on  the  exterior  surface  of  the  liquid  system,  are  alike  in  both 
cases. 

Employing  the  foregoing  notation,  we  obtain 


20  UNDERSHOT    WHEELS. 

1st.  For  the  mean  gain  in  the  projected  quantity  of  motion, 

V-  («'-»); 

y 

2d.     For  the  impulses  of  the  weight  and  pressures  together, 
also  taken  in  horizontal  projection,  %  Tl  ~b  6  (Aa  —  A'2)  ;  to  these 
impulses  is  to  be  added  that  produced  by  F,  or  —  F  d,  to  have 
the  sum  of  the  projected  impulses. 
We  will  then  have 


t/ 
whence  we  obtain 

F  =  -  (v  -  v')  -  6  n  5  f  (A/3-A2). 

The  forces  of  which  F  is  the  resultant  in  horizontal  projec- 
tion are  exerted  in  a  contrary  direction  by  the  water  on  the 
wheel,  at  points  whose  vertical  motion  is  nearly  null,  and 
whose  horizontal  velocity  is  approximately  v'.  The  wheel  will 
then  receive  from  these  forces,  in  each  unit  of  time,  a  work 
F  v',  which  represents  the  dynamic  effect  Te  to  within  a  slight 
error.  So  that 


'(h"  -  A3). 


Moreover  we  have 

P  = 
whence 


T.  =     o'  (o  -«/)- 


h       v' 
or  finally,  observing  that  p  =  — 


UNDERSHOT     WHEELS.  21 

In  order  that  this  formula  should  be  tolerably  exact,  the  depths 
h  and  h'  must  be  quite  small,  without  which  the  floats  would 
make  an  appreciable  angle  with  the  vertical  at  the  moment 
they  leave  the  water  ;  the  velocity  of  the  points  at  which  are 
applied  the  forces,  whose  resultant  is  F,  could  no  longer  be  con- 
sidered horizontal,  as  heretofore,  and  a  resistance  due  to  the 
emersion  of  the  floats  would  be  produced,  on  account  of  the 
liquid  uselessly  raised  by  them.  The  water  must  also  be  con- 
fined a  sufficiently  long  time  to  assume  the  velocity  vf. 

We  can  consider  in  formula  (1),  v  and  h  as  fixed  data,  and 
seek  the  most  suitable  value  of  the  velocity  vf  of  the  floats,  to 

vf 
make  Te  a  maximum.     For  this  purpose,  if  we  place  —  =  a?, 

v9 
Te  =  A  P  <r  —  ,  formula  (1)  becomes 


and  we  must  choose  a?  so  as  to  make  A  a  maximum.     This 
value  will  be  obtained  by  taking  the  value  which  reduces  the 

d  A 

first  differential  co-efficient  -^  —  •  to  zero,  which  gives  the  equa- 

tion 


The  value  of  x  deduced  depends  on  —^-\    we    find  approxi- 
mately 


For 

Sr  =  0.00 

x  =  0.500 

A  =  0.500 

V 

0.05 

0.553 

0.431 

0.10 

0.595 

0.373 

•n 

1 

g  A  /l 

\ 

22  UNDERSHOT   WHEELS. 

x  =  0.50,  A  =  0.50  ;  and  recollecting  that  the  maximum  value 

of  A  is  very  much  changed,  even  for  small  values  of  T*     Be- 

v 

sides,  experiment  does  not  show  that  there  is  any  use  in  taking 
x  greater  than  0.50,  as  we  have  just  found ;  it  would  rather 
show  a  ratio  of  v'  to  v  of  about  0.4,  which  obtains  probably 
because  of  the  motion  of  the  wheel  being  too  swift  to  allow  the 
liquid  to  pass  completely  from  the  velocity  v  to  v',  during  the 
time  that  it  remains  between  the  floats  :  a  portion  of  the  water 
passes  without  producing  its  entire  dynamic  effect,  and  the 
formula  (1)  ceases  to  conform  to  fact. 

Smeaton,  an  English  engineer,  made  some  experiments,  in 
1759,  on  a  small  wheel,  Om.609  in  diameter,  having  plane  floats. 
This  wheel  was  enclosed  in  a  race  having  a  flat  bottom,  which 
is  a  defect,  because  the  intervals  included  between  the  floats 
and  the  race  are  never  completely  closed.  The  weight  P  varied 
from  Ok.86  to  2.84  kilogrammes.  In  each  experiment  the  work 
transmitted  to  the  wheel  was  determined  by  the  raising  of  a 
weight  attached  to  a  rope  which  was  wound  around  the  axle. 

W 

The  most  suitable  value  of  a?  =  —  was  thus  found  to  vary  be- 
tween 0.34  and  0.52,  the  mean  being  0.43.  The  number  A, 
comprised  between  0.29  and  0.35,  had  for  a  mean  value  about  £. 

We  can  then  take  definitively  the  ratio  —  =  0.4,  as  found  by 

v 

experiment.     The  expression  for  A  becomes  then 

A  =  0.48 -2.1^, 

tf ' 

which,  for  ^  =  0.05  and^  =  0.10,  gives^the  numbers 

A  =  0.375  and  A  =  0.27, 
very  nearly  those  found  by  Smeaton. 


UNDERSHOT    WHEELS.  23 

A  few  remarks  remain  to  be  made,  to  which  it  would  be  well 
to  pay  attention  in  practice. 

1st.  It  is  well,  as  far  as  possible,  to  have  the  depth  h  say 
from  Om.15  to  Om.20.  Too  small  a  thickness  of  the  stratum  of 
water  which  impinges  on  the  wheel  would  give  a  relatively  ap- 
preciable importance  to  the  unavoidable  play  between  the 
wheel  and  the  race,  a  play  which  results  in  pure  loss  of  the 
motive  water.  Too  great  a  thickness  has  also  its  inconve- 
niences :  for  from  the  relation  j-  —  — ,  it  follows  that  for  —.  = 

h  vn  v' 

0.4  and  h  =  Om.20,  we  will  then  have 

*-!£=«••«» 

the  floats  will  then  be  immersed  Om.50,  adopting  the  thickness 
of  Om.20,  and  were  they  more  so,  they  would  meet  with  consi- 
derable resistance  in  leaving  the  water,  as  we  have  already  said. 
It  is  necessary  then  that  h  be  neither  too  large  nor  too  small : 
the  limits  of  Om.15  to  Om.20  are  recommended  by  M.  Belan- 
ger. 

2d.  We  should  avoid  as  much  as  possible  losses  of  head  dur- 
ing the  passage  of  the  water  from  the  basin  up-stream  to  its 
arrival  at  the  section  C  B  near  the  wheel,  losses  which  result 
in  a  diminution  of  effective  delivery  (No.  2).  In  order  to  give 
it  the  shape  of  a  thin  layer,  from  Om.15  to  Om.20  in  thickness, 
the  water  is  made  to  flow  under  a  sluice  through  a  rectangular 
orifice :  care  should  be  taken  to  avoid  abrupt  changes  of  direc- 
tion between  the  sides  of  this  orifice  and  the  interior  of  the 
dam,  in  order  to  avoid  a  contraction  followed  by  a  sudden 
change  of  direction  of  the  threads,  as  in  cylindrical  orifices. 
The  sluice  should  be  inclined  (as  in  Fig.  4),  in  order  to  leave 
but  a  small  distance  between  the  orifice  and  the  wheel,  which 
will  diminish  the  loss  of  head  produced  by  the  friction  of  the 


24:  UNDERSHOT   WHEELS. 

water  on  the  portion  of  the  race  M  C,  through  which  the  water 
reaches  the  wheel. 


FIG.  4. 

3d.  The  water  leaves  the  wheel  at  E  F  with  a  velocity  v'  = 
0.4  v,  in  the  shape  of  a  horizontal  band  of  parallel  threads. 
If,  in  order  to  flow  into  the  tail  race  through  a  section  G  K, 
where  its  velocity  will  be  sensibly  zero,  it  had  to  undergo  no 
loss  of  head,  there  would  be  between  E  F  and  G  K  a  negative 

•y/a  -y2 

head,  the  absolute  value  of  which  would  be  ^— ,  or  0.16  ,r— ;  that 

2  g  %g 

v* 

is,  the  point  G  would  be  at  a  height  0.16  ^-  above  E,  since  the 

*9 

piezometric  levels  at  E  F  and  G  H  may  be  confounded  with 
those  of  the  points  E  and  G.  It  is  n<  t  possible  so  to  arrange 
every  part  that  all  loss  of  head  shall  be  suppressed  between 
E  F  and  G  K  ;  but  these  losses  are  much  diminished  by  means 
of  a  plan  first  recommended  by  M.  Bel  anger.  The  bottom  of 
the  race,  beyond  the  circular  portion  A  D,  presents  a  slight 
slope,  for  a  distance  D  I  =  1  or  2  metres ;  thence  it  connects 
with  the  tail  race  by  a  line  I  K,  having  an  inclination  from  Om.07 
of  a  metre  to  Om.10  for  each  metre  in  length  ;  the  side- walls  are 
prolonged  for  the  same  distance,  either  keeping  their  planes 
parallel,  or  very  gradua^y  spreading  outwards,  but  never  ex- 


UNDERSHOT   WHEELS.  25 

ceeding  3  or  4  degrees.     The  point  D  is  placed  at  a  height 

h'  +  |,  0.16  ~  (*),  or  2.5  A  +  0.11  ~  below  the  level  of  the 
^  9  ^  ff 

water  in  the  tail  race.  From  this  the  following  effects  take 
place :  the  water  overcomes  the  difference  of  level  between  E 

and  G,  or  the  height  0.11  — ,  in  virtue  of  its  velocity  0.4  v, 

either  by  a  surface  counter-slope,  or  by  a  sudden  change  of 
level  with  a  counter-slope,  so  that  the  head  lost  reduces  to 

0.05 /, 
2<7 

In  many  wheels  this  precaution  has  been  neglected,  and  the 
level  of  the  tail  race  has  been  placed  at  the  same  height  as  the 
point  E,  and  sometimes  even  below  it.  We  will  now  show 
that  a  loss  of  effective  delivery  is  thus  produced.  For  this  pur- 
pose let  us  see  what  must,  with  the  above-described  arrange- 
ment, be  the  position  of  the  race  relative  to  the  pond,  and  the 
expression  for  the  effective  delivery.  In  view  of  simplifying 
this  investigation,  we  will  admit  that  the  lines  MA,  D  F, 
slightly  inclined,  are  a  portion  of  the  same  horizontal.  In  the 
hypothesis  of  no  loss  of  head  up  to  B  C,  the  velocity  v  would 
be  due  to  the  height  z  of  the  level  N  in  the  pond,  above  the 
highest  thread  of  the  fluid  vein  thrown  on  the  wheel ;  but,  on 

2 

*  The  co -efficient  o   is  simply    assumed :    in  replacing  it  by  unity  the 

water  would  no  longer  be  able  to  attain  the  level  of  the  point  G,  since  this 
would  require  a  loss  of  head  which  would  be  null  in  the  interval  between 
E  F  and  G-  K ;  consequently  the  water  in  the  lower  race  would  probably 
drown  the  floats  and  impede  their  motion.  It  is  for  the  purpose  of  avoiding 
such  an  inconvenience  that  the  number  in  question  is  taken  less  than  unity  ; 

1  vn  tf 

the  value  assumed  gives  a  disposable  head  expressed  by  ^  g— ,  or  O.OSg-,  to 

counterbalance  the  loss  of  head  of  the  liquid  molecules  after  escaping  from  the 
wheel,  and  to  secure  for  the  wheel  their  free  discharge. 


26  UNDERSHOT   WHEELS. 

account  of  the  losses  of  head,  we  give  z  a  co-efficient  of  reduc- 
tion, which  we  will  take  (for  want  of  precise  data)  equal  to 
0.95  ;  that  is,  we  will  write 

£-  =  0.95  z. 
whence 


~  0.95'  2  g 
The  distance  from  the  bottom  M  A  to  the  level  N"  will  then  be 

h  +  Tt-Qft  y—  ;  this  can  also  be  expressed  in  another  way,  thus : 

V  ' 

H  +  2.5  h  +  |,  0.16^-, 

by  calling  H  the  total  head,  or  the  difference  in  height  between 
G  and  N".     Hence 


and  consequently 


a  relation  which  gives  -y,  for  any  given  head,  when  the  value  of 
h  has  been  determined.  From  this  we  can  deduce  z  and  h  +  z, 
which  is  sufficient  to  determine  the  position  of  the  race.  The 
dynamic  effect  Te,  from  what  we  have  just  seen,  will  be 

tf  vf 

A  P     "'  ° 


V 

substituting  for  ^—  its  value,  this  relation  becomes   Ta  =  P 

[  0.48  x  1.057   (  H  +  I  h  )  -  2.1-  ^  A  ]    =  P  (0.507  H  - 
0.289  h) 


UNDERSHOT   WHEELS.  27 

T 

The  effective  delivery  p-4j  will  then  be  expressed  in  round 


numbers  by 


—  0.50  —  O.o  TT  5 


p   TT    - 

for  A  =  Om.20  and  H  included  between  1  metre  and  2  metres, 
it  would  vary  from  0.44  to  0.47. 

Now  let  us  suppose  that,  without  changing  the  head  H,  we 
wish  to  place  the  level  of  the  tail  race  below  E,  or,  at  most,  on 
the  same  level ;  it  is  plain  that  we  will  have  to  raise  the  bottom 
of  the  race.  Then  v  will  diminish,  and  h  will  have  to  increase 
in  order  that  the  expenditure  P  may  remain  the  same;  for 
these  two  reasons  Te  will  diminish,  as  the  above  formula  (2) 
shows.  For  example,  supposing  that  the  tail  race  is  at  the 

height  of  E,  the  equation  that  determines  - —  will  become 


whence  we   derive  successively,  regard  being  had  to  equa- 
tion (2), 


Te  =  P  [0.48  x  0.95  (H  +  |  A)  -  1.05  A]  =  P  (0.456  H  - 


0.375  h\ 

and  in  round  numbers 


=0.45-  cum, 


The  data  h  =  Om.20  and  H  =  1  metre  would  give  a  effective  de- 
livery of  0.38  instead  of  0.44 ;  with  H  =  2  metres,  we  would 
obtain  an  effective  delivery  of  0.41,  whilst  we  had  found  0.47. 


28  UNDERSHOT   WHEELS. 

There  would  be  a  much  more  marked  falling  off,  if  we  supposed 
the  bottom  of  the  tail  race  on  the  same  level  as  the  bottom  of 
the  portion  preceding  it,  as  was  the  manner  of  constructing  the 
race  formerly. 

The  arrangement  of  the  channel  through  which  the  water 
flows  off,  which  we  have  mentioned  as  by  M.  Belanger,  can  be 
advantageously  employed  in  all  systems  of  hydraulic  wheels 
from  which  the  water  flows  with  a  sensible  velocity,  in  the 
form  of  a  horizontal  current,  with  parallel  threads.  The  ris- 
ing of  the  surface  of  this  current  taking  place  beyond  the 
wheel,  this  latter  will  experience  the  same  action  from  the 
water,  if  everything  is  similarly  arranged  from  the  head  race 
to  the  outlet  of  the  wheel.  With  the  canal  in  question,  the 
level  rises,  instead  of  remaining  the  same  or  falling;  we 
then  obtain  the  same  action  on  the  wheel  with  a  less  head, 
and  consequently  we  can  have  a  greater  effect  the  head  remain- 
ing the  same. 

However,  we  see  by  formula  (3)  that  the  effective  delivery 
of  these  wheels  never  reaches  0.50,  in  spite  of  all  possible  pre- 
cautions ;  this  system  is,  then,  not  comparatively  as  good  as 
those  which  we  are  now  going  to  take  up. 

5.  Wheels  arranged  according  to  Ponceletfs  method. — The 
principal  cause  of  loss  of  work  in  the  undershot  wheel  with 
plane  floats  is  the  sudden  change  from  the  velocity  v  to  v', 
twice  and  a  half  less,  which  necessarily  produces  in  the  liquid 
a  violent  disturbance.  From  this  disturbance  proceed  great 
inner  distortions  and  a  negative  work  produced  by  viscosity, 
all  of  which  diminishes  the  dynamic  effect.  The  water  also 
possesses  a  great  velocity  of  exit,  which  is  at  best  only  partially 
turned  to  account.  General  Poncelet  proposed  to  avoid  these 
inconveniences,  continuing,  however,  to  preserve  to  the  wheel 
its  special  character,  which  is  rapid  motion ;  that  is,  he  has 


UNDERSHOT   WHEELS.  29 

endeavored  to  fulfil  for  the  undershot  wheel  the  two  general 
conditions  of  a  good  hydraulic  motor,  viz. :  the  entrance  of  the 
water  without  shock,  and  its  exit  without  velocity.  To  this 
end  he  has  contrived  the  following  arrangements : 

The  bottom  of  the  head  race,  which  is  sensibly  horizontal,  is 
joined  without  break  to  the  flume,  the  profile  of  which  is  com- 
posed of  a  right  line  of  ^ ,  followed  by  a  curve. 

The  right  line  forms  a  slope  near  the  wheel,  and  its  prolon- 
gation would  be  tangent  to  the  outer  circumference  of  this  lat- 
ter ;  it  ends  at  the  point  at  which  the  water  begins  to  enter 
the  wheel.  The  curved  portion  is  composed  of  a  special  curve, 
to  the  shape  of  which  we  will  return  presently,  and  which 
stops  at  the  point  at  which  it  meets  the  exterior  circumference. 
Finally,  the  floats  are  set  in  a  cylindrical  portion  of  the  race, 
having  a  development  a  little  greater  than  the  interval  beweeri 
two  consecutive  floats,  and  terminated  by  an  abrupt  depression  ; 
this  depression  has  its  summit  at  the  mean  level  of  the  water 
down-stream ;  its  object  is  to  facilitate  the  discharge  of  the 
water  from  the  wheel. 

The  water  enters  the  race  under  a  sluice  inclined  at  an 
angle  of  from  30  to  45  degrees  with  the  vertical ;  the  sides  of 
the  orifice  are  rounded  off,  so  as  to  avoid  the  loss  of  head  analo- 
gous to  that  in  cylindrical  orifices. 

The  floats  are  set  between  two  rings  or  shrouds,  which  pre- 
vent the  water  from  escaping  at  the  sides ;  the  interior  space 
between  the  rings  is  somewhat  greater  than  the  breadth  of  the 
orifice  opened  by  the  sluice.  The  floats  are  curved  ;  they  in- 
tersect the  outer  circumference  of  the  crown  at  an  angle  of 
about  30  degrees,  and  are  normal  to  the  inner  circumference ; 
beyond  this,  their  curvature  is  a  matter  of  indifference.  There 
are  ordinarily  36  for  wheels  of  from  3  to  4  metres  in  diameter, 
and  48  for  those  from  6  to  7  metres. 


30  UNDERSHOT    WHEELS. 

The  exact  theory  of  this  wheel  it  is  almost  impossible  to 
explain  in  the  present  state  of  science.  It  is  simplified,  first, 
by  considering  the  mass  of  water  that  enters  between  two  con- 
secutive floats  as  a  simple  material  point,  which,  during  its 
relative  motion  in  the  wheel,  would  experience  no  friction. 
We  suppose,  moreover,  that  the  absolute  velocity  of  this  point 
is  in  the  direction  of  the  horizontal  that  touches  the  wheel  at 
its  lowest  point,  and  that  the  floats  are  so  put  on  as  to  be  tan- 
gent to  the  exterior  circumference.  Now,  let  v  be  the  absolute 
velocity  of  the  water  on  striking  the  wheel,  and  u  the  velocity 
on  this  circumference ;  the  water  possesses  relatively  to  the 
wheel,  at  the  moment  of  entering  the  floats,  a  horizontal  velo- 
city v  —  u,  in  virtue  of  which  it  takes  up  a  motion  towards  the 
interior  of  the  basin  formed  by  two  consecutive  floats  If  we 
liken,  during  this  very  short  relative  motion,  the  motion  of  the 
floats  to  a  uniform  motion  of  translation  along  the  horizontal, 
the  apparent  forces  will  reduce  to  zero,  so  that  the  small  liquid 
mass  that  we  have  spoken  of  will  ascend  along  the  floats  to  a 

Ay ^y 

height  — ~ -,  on  account  of  its  initial  relative  velocity,  which 

is  gradually  destroyed  by  the  action  of  gravity.  Then  this 
mass  descends,  and  again  takes  up  the  same  relative  velocity, 
v  —  u,  when  on  the  point  of  leaving  the  float ;  but  this  relative 
velocity  will  be  in  a  direction  contrary  to  the  preceding,  and, 
consequently,  also  in  a  direction  contrary  to  the  velocity  u  of 
the  floats.  The  absolute  velocity  of  the  water  on  leaving  will 
then  be  equal  to  the  difference  between  u  and  v  —  u,  or  2u  —  v  ; 

we  see  that  it  will  be  zero  if  we  have  u  =  -  v,  that  is,  provided 

that  the  velocity  at  the  outer  circumference  of  the  wheel  is  half 
that  of  the  water  in  the  supply  channel. 


UNDERSHOT   WHEELS.  31 

We  could  thus  realize  the  two  principal  conditions  for  a 
good  wheel.  But,  as  M.  Poncelet  has  observed,  such  favorable 
circumstances  are  found  by  no  means  in  practice. 

The  water  cannot  enter  the  wheel  tangent  to  its  circumfer- 
ence, as  we  have  here  supposed.  In  fact,  let  us  call  ds  the 
length  of  an  element  of  this  circumference  immersed  in  the 
current,  b  the  breadth  of  the  wheel,  p  the  angle  formed  by  ds 
and  the  relative  velocity,  w,  of  the  water  referred  to  the  wheel ; 
there  will  enter  during  a  unit  of  time,  through  the  surface  bds, 
a  prismatic  volume  of  liquid  having  for  a  right  section  Ids.  sin 
/s,  and  a  length  w  ;  in  other  words,  a  volume  Ids.w  sin  /3,  a 
quantity  that  reduces  to  zero,  for  p=Q.  Hence  w  must  inter- 
sect the  circumference  at  a  certain  angle  which  cannot  be  zero  ; 
besides,  we  must  make  it  as  small  as  possible,  in  order  that,  at 
the  point  of  exit,  the  relative  velocity  of  the  liquid  and  the 
velocity  of  the  floats  may  be  sensibly  opposite,  and  give  a  resul- 
tant zero ;  on  the  other  hand,  it  must  not  be  so  small  as  to 
make  the  entrance  of  the  water  difficult  or  impossible.  It  is 
in  order  to  reconcile  these  two  contradictory  conditions  that 
the  angle  £  has  been  fixed  at  30  degrees,  which  is  also  that  made 
by  the  floats  with  the  exterior  circumference,  since  the  threads 
must  enter  in  the  direction  tangent  to  the  floats.  But  then 
the  absolute  velocity  of  the  water  is  no  longer  zero  on  leaving 
the  wheel,  for  its  two  components  are  no  longer  following  the 
same  right  line,  but  make  an  angle  of  180°— 30°  or  150°. 
Now,  supposing  v  —  u  =  u^  the  resultant  vf  would  have  for  a 
value  %u  cos  75°,  or  v  cos  75°,  or  finally,  0.259-y  /  this  resultant 
lies  furthermore  in  the  direction  of  the  bisecting  line  of  the 
angle  between  u  and  v  —  u ;  that  is  to  say,  it  is  almost  vertical, 
and  consequently  it  is  impossible  to  turn  it  to  account  by 
means  of  a  counter-slope ;  its  effect  is  destroyed  in  producing  a 
disturbance  in  the  lower  portion  of  the  canal,  whence  there 


32  UNDERSHOT   WHEELS. 


I 

results  a  negative  work  equal  to  P  —  ,  P  being  the  expenditure 

fi" 

of  the  head.     It  is  then  a  loss  of  head  expressed  by  —  ,  or 

0.067-1-  . 
2<7 

As  a  cause  of  loss  we  may  still  mention  the  friction  of  the 
water  against  the  race  and  against  the  floats.  Another  very 
serious  objection  to  the  above-mentioned  theory  is  that  the 
liquid  molecules  do  not  move  as  though  they  were  entirely 


isolated  ;  when  one  of  them,  having  reached  the  height 

between  the  floats,  is  ill  its  descent,  another  has  just  entered, 
and  it  is  not  clear  that  no  sensible  disturbance  in  the  motion 
of  the  molecules  follows. 

On  account  of  all  these  reasons,  experiment  indicates  only 
an  effective  mean  work  of  0.60  for  these  wheels.  Nevertheless, 
the  improvement  is  very  great  on  the  old  undershot  wheels 
with  plane  floats,  whose  effective  work  was  scarcely  0.25  or 
0.30,  and  could  never  reach  the  limit  0.50.  As  to  the  most 
favorable  ratio  between  the  velocities  w  and  v,  experiment 

gives  it  0.55  instead  of  -. 

It  has  been  observed  that  the  straight  portion  of  the  race  was 
followed  by  a  curve;  the  following  condition  determines  its 
form.  The  relative  velocity  of  the  water  v  —  u  at  its  entrance, 
equal  to  that  which  exists  at  its  exit,  is  also  the  same  as  the  cir- 
cumference velocity  u  ;  then  the  direction  of  the  absolute  velo- 
city, resulting  from  these  two  velocities,  is  on  the  line  bisecting 
the  angle  formed  by  these  two,  and  since  the  angle  between 
the  tangent  to  the  exterior  circumference  and  the  relative 
velocity  is  30  degrees,  that  between  the  same  tangent  and  the 
absolute  velocity  will  be  15  degrees.  All  the  threads,  then, 


UNDERSHOT    WHEELS.  33 

must  make  an  angle  of  15  degrees  with  the  circumference. 
To  deduce  from  this  the  shape  of  the  bottom,  let  us  assume 
that  every  normal  to  the  bottom  is  at  the  same  time  normal  to 
all  the  threads  that  it  intersects.  Let  A  then  (Fig.  5)  be  the 
point  of  entrance  of  a  thread,  the  absolute  velocity  A  v  of  which 
makes  an  angle  of  15 
degrees  with  the  tan- 
gent AM/  the  perpen- 
dicular BAB' to  Av 
is  a  normal  to  the 
bottom  of  the  race. 
At  the  same  time,  if 
we  draw  the  radius 
AO,theangleOAB 
will  be  equal  to  vAy,  as  their  sides  are  perpendicular ;  hence 
O  A  B'  =  15  degrees.  The  perpendicular  O  B'  let  fall  from  the 
centre  O  to  the  prolongation  of  B  A,  has  then  a  constant  value 
equal  to  A  O  sin  15°.  Consequently,  all  the  normals  to  the 
bottom  of  the  race  are  tangent  to  the  same  circle,  having  a 
radius  O  B' ;  the  curve  C  B  D  is  then  the  involute  of  which  this 
circle  is  the  evolute.  It  is  terminated  at  one  end  by  the  cir- 
cumference OA,  at  the  point  D,  and  at  the  other  at  a  point  C, 
such  that  the  normal  C  E,  taken  as  far  as  the  circumference 
O  A,  may-.have  a  length  equal  to  the  thickness  of  the  fluid 
stratum.  This  thickness,  moreover,  varies  with  the  head  ;  it 
must  be,  according  to  experiment,  from  Om.20  to  Om.30  for 
heads  below  lm.50,  and  may  be  diminished  to  about  Om.10  for 
heads  of  more  than  2  metres. 

The  water  rises  in  the  float  to  a  height  ^— — '   =  -  — , 
which  differs  little  from  --  H,  calling  H  the  height  of  the  head. 


34:  UNDERSHOT   WHEELS. 

The  distance  between  the  two  circumferences  that  limit  the 
float  should  be  at  least  -  H  ;  in  order  more  certainly  to  avoid 

the  possibility  of  the  water  still  possessing  any  relative  velocity 
on  reaching  the  extremity  of  the  floats,  which  would  give  rise 
to  a  spirting  of  the  water  to  the  interior  of  the  wheel,  M.  Pon- 

celet  has  advised  increasing  this  advance  to  -  H. 

o 

6.  Paddle  wheels  in  an  unconfined  current. — These  wheels 
are  placed  in  a  current  whose  section  has  a  breadth  much 
greater  than  that  of  the  wheel ;  frequently  they  are  supported 
by  boats,  and  are  called  hanging  wheels.  It  being  impossible 
to  calculate  theoretically  the  dynamic  effect  of  the  current  on 
these  wheels,  we  will  be  satisfied  with  the  following  general 
ideas. 

The  horizontal  force  F  which  the  wheel  exerts  on  the  liquid 
being  constant,  its  impulse  in  the  unit  of  time  has  numerically 
the  same  value  as  its  intensity.  If,  then,  owing  to  this  impulse, 
a  mass  m  of  water  passes,  in  a  second,  from  the  velocity  v  of 
the  current  to  the  velocity  u  of  the  wheel,  we  shall  have 

F  =  m  (v  —  u\ 

and  consequently  the  work  produced  on  the  wheel  in  the  same 
time,  sensibly  equal  to  the  dynamic  effect  of  the  current,  will 
be 

F  u  =  m  u  (v  —  u). 

Moreover,  General  Poncelet  supposes  that  the  mass  m  must 
be  proportional  to  v,  and,  furthermore,  it  is  quite  natural  to 
admit  that  it  is  proportional  to  the  area  S  of  the  immersed 
portion  of  the  floats.  He  then  places,  calling  B  a  constant 
co-efficient  and  n  the  weight  of  a  cubic  yard  of  water, 

m  =  ESv  — 

9 


UNDERSHOT   WHEELS.  35 

whence  we  have 


9 

This  formula  has  been  found  quite  true  by  experiment,  in 
taking  B  =  0.8. 

When  v  is  given,  the  maximum  state  of  F  u  corresponds  to 
u  =  v  —  u,  since  the  sum  of  these  two  factors  is  constant  :  from 

this  we  deduce  u  =  -  v,  as  we  found  for  the  two  wheels  pre- 

2i 

viously  discussed.     Experiment  indicates  the  ratio  -  =  0.4  as 

being  the  most  suitable;  this  ratio  only  changes  very  slightly 
the  theoretical  maximum,  the  value  of  which  is 


The  depth  of  the  floats  must  be  from  -  to  -  the  length  of 

the  radius.  Flanges  placed  on  the  edge,  on  the  side  which  re- 
ceives the  shock  of  the  water,  will  increase  the  mutual  action. 
The  diameter  is  ordinarily  from  4  to  5  metres,  and  the  floats 
are  12  in  number. 

3 


BREAST  WHEELS. 


7.  Wheels  set  in  a  circular  race,  called  breast  wheels. — These 
wheels  resemble  very  closely,  in  their  construction,  the  under- 
shot wheels  noticed  in  No.  4 ;  but  one  essential  difference,  which 
can  produce  a  great  change  in  their  effective  delivery,  con- 
sists in  the  method  of  introducing  the  water ;  this  no  longer 
enters  the  wheel  at  its  lower  portion,  but  at  a  slight  depth  only 
below  the  axle — that  is  to  say,  on  the  side.  The  questions  now 
to  be  successively  examined  will  be,  1st,  the  most  suitable  ve- 
locity of  the  wheel ;  2d,  the  method  of  introducing  the  water ; 
3d,  the  situation  of  the  race  as  regards  the  upper  end  and 
tail  race ;  4th,  the  manner  of  determining  the  dynamic  effect. 
Finally,  several  practical  ideas  will  be  put  forth  on  the  subject 
of  this  wheel. 

(a)  To  determine  the  velocity  of  the  wheel. — Let  Q  be  the 
volume  that  the  head  expends  in  a  second ; 

b  the  breadth  of  the  wheel,  which  is  the  same  as  that  of  the 
sluice ; 

h  the  depth  of  the  immersed  portion  of  the  floats  directly 
beneath  the  axle; 

c  the  thickness  of  the  floats ; 

C  their  distance  apart,  measured  between  their  axes  along 
the  exterior  circumference ; 

R  the  radius  of  this  circumference ; 

u  the  velocity  of  any  one  of  its  points. 


BREAST   WHEELS.  37 

The  water  comprised  between   any  two  consecutive  floats, 
directly  beneath  the  axle,  will  have  for  its  depth  A,  for  its  mean 

breadth  C  (l  —  -=\  —  c,  and  I  for  its  thickness  parallel  to  the 
axis  of  the  wheel ;  its  volume  is  then  h  I C  (l  —  — ^  —  ^- V  and, 


u 


supposing  that  it  expends  during  one  second  a  number  —  of 

C 

these  volumes,  we  shall  have 


h  c 

In  practice  — -^  and  —  are  always  small  fractions,  the  sum  of 
2i  1C         \j 

which   rarely  exceeds   0.10 ;   we  would  then   have,  except  a 
slight  error, 

Q  =  0.9  h  I  u. 

Q  being  given,  we  can  satisfy  this  relation  by  assuming  u  and 
h  arbitrarily,  and  then  determining  b.     In  this  case,  we  would 

take  h  from  Om.15  to  Om.25,  u  =  lm.30,  and  finally,  I  =  ^^; 

y  fi  u 

the  expenditure  -^  per  metre  in  breadth,  expressed  by  0.9  A  u, 

would  then  vary  from  175  litres  to  about  270  litres. 

The  values  just  given  for  A  and  u  may  be  satisfactorily  ac- 
counted for  as  follows : 

The  calculation  of  the  losses  of  head  sustained  by  the  water, 
in  its  passage  from  the  head  race  to  the  tail  race,  shows,  as  we 
shall  see  further  on,  that  all  these  losses  increase  with  the  velo- 
city of  the  wheel.  We  should  naturally  be  inclined  to  make 
this  velocity  very  small,  in  order  to  increase  the  effective  deliv- 
ery. But  several  conditions  show  that  it  is  not  well  to  allow 


38  BREAST   WHEELS. 

the  wheel  to  move  too  slowly.     In  fact,  we  see  that  if  u  were 

very  small,  the  product  hb  =  -  -  —  would  be  large,  and  one  of 

y  it 

the  two  dimensions  5  or  h  would  have  to  be  quite  large.  Now 
a  great  breadth  b  would  give  us  a  heavy  wheel,  expensive  to 
put  up,  losing  a  great  deal  of  work  by  the  friction  of  the  axle  ; 
a  great  value  of  h  would  produce  inconveniences  already  men- 
tioned in  speaking  of  the  undershot  wheel  (No.  4).  Finally,  it  is 
well  to  have  the  wheel  perform,  up  to  a  certain  point,  the  func- 
tions of  a  fly-wheel  for  the  machinery  that  it  sets  in  motion, 
which  is  another  reason  for  allowing  it  to  retain  a  certain 
velocity.  The  value  u  =  lm.30  has  been  given  by  experiment. 
As  regards  the  depth  A,  it  is  well,  on  account  of  the  unavoida- 
ble play  between  the  wheel  and  the  race,  that  it  should  not 
descend  much  below  Om.15,  in  order  not  to  lose  too  great  a  pro- 
portion of  water. 

But  it  sometimes  happens  that,  in  taking  u  —  lm.30,  and  A 
within  the  limits  above  mentioned,  we  arrive  at  a  great  value 
for  5,  or  a  value  that  goes  beyond  a  fixed  limit,  either  on  ac- 
count of  local  circumstances,  or  through  economy  in  construc- 
tion. We  are  then  obliged  to  increase  A  or  u.  It  is  not  well 
to  have  A  greater  than  Om.45  or  Om.50,  and  when  this  limit  is 
reached,  we  must  then  begin  to  increase  u. 

For  example  :  let  Q  =  Om.50.     Taking  u  =  lm.30,  A  =  Om.20, 

we  would  have  I  =  --  ^-  —  2m.14,  a  value  that  in  general  is 

quite  admissible.  But  if,  by  reason  of  particular  circum- 
stances, we  could  not  exceed  a  breadth  of  Om.71,  about  the 
third  of  2m.14,  we  should  first  increase  A,  bringing  this  up  to 


Om.50  ;  we  would  then  find  -M  ==  =  lm.56.    Or  else,  as  the 


BREA.ST    WHEELS. 


39 


velocity  lm.56  is  not  yet  very  great,  we  would  be  content  to 
take  h  —  Om.40,  which  would  give  u  =  lm.96. 

(b.)  Method  of  introducing  the  water. — As  has  already  been 
said  (No.  3),  in  order  to  diminish  as  much  as  possible  the  loss 
of  work  produced  by  the  introduction  of  the  water  on  the 
wheel,  we  must  so  arrange  matters  as  to  have  a  small  relative 
velocity  of  the  water  at  its  point  of  entrance,  or,  when  that 
cannot  be  obtained,  the  relative  velocity  must  be  tangent  to 
the  first  element  of  the  floats,  and  the  water  must  move  to  the 
interior  of  the  wheel  without  shock,  merely  gliding  along  the 
solid  sides. 

If  the  wheel  be  moving  slowly — that  is,  if  the  velocity  at  the 
circumference  be  about  lm.30  per  second,  we  should  let  the 
water  in  by  the  means  indicated  in  the  following  figure. 


Pro.  6. 


The  flume  A  B,  constructed  of  masonry,  is  prolonged  by  a 
piece  of  cast  iron  B  C,  called  a  swan's  neck,  or  guide  bucket ; 
A  B  C  is  the  arc  of  a  circle  nearly  coincident  with  the  exterior 
circumference  of  the  wheel,  leaving  the  slightest  amount  of 


40  BREAST   WHEELS. 

play.  A  sluice  D,  furnished  at  its  upper  extremity  with  a 
small,  rounded  metallic  appendage  E,  can  slide  along  the 
guide  bucket  while  resting  on  it ;  a  system  of  two  racks  con- 
nected by  cog-wheels  allows  this  sluice  to  be  raised  to  any  con- 
venient point.  The  metallic  appendage  E  forms  the  sole  of  a 
weir  over  which  the  water  flows  to  get  to  the  wheel ;  this  sole 
must  be  from  Om.20  to  Om.2T  below  the  level  of  the  pond. 

The  sluice  being  thus  very  near  the  wheel,  the  absolute 
velocity  of  the  water  at  its  entrance,  due  only  to  the  slight  fall 
which  takes  place  in  the  surface  of  the  pond,  will  consequently 
be  small ;  and  as  the  wheel  itself  moves  slowly,  the  relative 
velocity  will  be  moderate,  and  the  disturbance  produced  hardly 
perceptible.  We  see  that,  in  order  to  attain  this  end,  we  should 
reduce  the  head  over  the  sole  of  the  weir,  for  in  the  contrary 
case  the  velocity  of  flow  would  have  a  greater  or  less  value ;  we 
should  not,  moreover,  carry  the  reduction  too  far,  in  order  that 
the  loss  of  water  between  the  wheel  and  flume  may  not  be  too 
great.  The  limits  Om.20  to  Om.27  fulfil  this  double  condition 
quite  well;  they  correspond  very  nearly  to  the  limits  Om.15  and 
Om.25  above  mentioned  for  the  depth  h  of  the  current  just 
beneath  the  axle ;  for  a  weir  without  lateral  contraction  can 
yield  about  180  litres  per  metre  of  breadth,  with  a  head  of 
Om.20  above  the  sole,  and  280  litres  with  a  head  of  Om.27. 

The  rounded  metallic  appendage  E  has  for  its  object  to 
diminish  the  contraction  of  the  sheet  of  water  flowing  over  the 
weir,  and  thus  for  the  same  head  to  yield  more  water:  the  head 
and  velocity  of  flow  are  then  less  for  a  given  yield,  which 
diminishes  the  disturbance  of  the  liquid  on  entering.  The 
piece  E  may  also  serve  to  direct  the  vein  so  as  not  to  intersect 
the  exterior  circumference  of  the  wheel  at  too  great  an  angle ; 
this  is  as  it  should  be,  for,  all  other  things  being  equal,  the 
relative  velocity  of  the  water  increases  with  the  angle  in  ques- 


BREAST   WHEELS.  41 

tion,  as  the  following  considerations  show,  although  they  more 
particularly  apply  to  a  different  case. 

Let  us  now  pass  to  the  case  of  rapidly  moving  wheels.  The 
velocity  u  of  the  wheel  at  its  circumference  has  been  fixed  as 
has  been  shown  ;  but  we  still,  in  order  to  lessen  the  loss  due  to 
the  introduction  of  the  water,  dispose  of  the  absolute  velocity 
v  of  this  latter,  in  intensity  and  direction.  Let,  at  the  point 
of  entrance,  M  (Fig.  7),  M  U  be  the  velocity 
u  of  the  wheel,  M  Y  the  absolute  velocity  v 
of  the  water,  7  the  angle  formed  by  these  two 
right  lines.  The  line  M  W,  equal  and  paral- 
lel to  U  Y,  will  be  the  velocity  w  of  the  water 
relatively  to  the  wheel.  The  first  point  is  to 
make  w  as  small  as  possible.  ]S~ow  we  should 
FIG  7  have  w  =  <9,  if  7  were  zero  and  v  equal  to  u  • 

but  it  is  not  possible  for  the  water  to  enter 
without  any  relative  velocity,  and  at  an  angle  zero.  One 
thread  for  which  7  may  have  the  value  0,  would  be  tangent  to 
the  exterior  circumference  of  the  wheel ;  but  it  is  evident  that 
the  other  threads,  placed  in  juxtaposition  parallel  to  this,  could 
not  fulfil  the  same  condition,  unless  the  total  thickness  of  these 
threads  taken  together  were  itself  zero,  which  cannot  be  admit- 
ted in  the  case  of  a  finite  expenditure.  We  have  seen  that  in 
Poncelet's  wheel  the  angle  7  was  taken  equal  to  15  degrees  ;  in 
the  breast  wheel  it  is  generally  taken  to  be  30  degrees,  in  order 
to  somewhat  facilitate  the  introduction  of  the  water.  Then, 
since  U  is  a  point  determined  as  well  as  the  direction  M,  the 
smallest  possible  value  of  IT  Y  =  w  would  be  the  perpendicular 

let  fall  from  U  on  M  Y,  or  u  sin  7,  or  finally  -  u  if  7  be  taken 

2 

equal  to  30° ;  v  would  have  for  a  corresponding  value  u  cos  7, 
or  0.866  v,  7  being  30°. 


42  BREAST   WHEELS. 

Still  these  values  are  not  those  taken  for  v  and  w.  It  is  well, 
in  fact,  to  have  the  first  element  of  the  floats  in  the  direction 
of  w,  in  order  to  diminish  the  disturbance  of  the  water  within 
the  wheel.  Now,  if  this  first  element  were  perpendicular  to 
M  Y,  it  would  make  an  angle  of  120°  with  M  U — that  is,  with 
the  circumference  of  the  wheel,  and  consequently  an  angle  of 
30°  with  the  radius  through  M.  When  this  radius,  by  the  act 
of  revolving,  reaches  the  vertical,  immediately  below  the  axle, 
the  first  element  of  the  float  will  still  be  inclined  at  an  angle 
of  30°  with  the  vertical,  and  it  would  be  still  greater  at  the 
point  of  emersion,  which  would  obstruct  the  motion  of  the 
wheel.  On  this  account  it  is  better  that  the  relative  velocity 
M  W  should  pass  through  the  axis  of  rotation  and  be  perpen- 
dicular to  M  U ;  the  parallelogram  of  velocities  is  then  repre- 
sented by  the  rectangle  MTJV'W,  in  which  M  U  =  u, 
M  W '  —  w,  M  Y '  =  v.  The  angle  7  still  preserving  its  value 
of  30°,  we  have  these  relations : 

u  i  KK 

V  =  ^30°  =  1'55  M' 
w  =  u  tang  30°  =  0.577  u. 

The  intensity  of  v  and  its  angle  with  u  being  henceforth  de- 
termined, it  remains  to  be  seen  how  in  practice  these  two  con- 
ditions can  be  realized.  To  obtain  them,  we  will  first  deter- 

v* 
mine  the  height  -  — ;  we  will  increase  it  a  little,  say  by  one- 

2  9 

tenth,  in  order  to  compensate  approximately  for  the  losses  of 
head  sustained  by  the  liquid  between  the  head  race  and  the 

v* 
point  of  entrance,  and  1.1  —  will  be  the  depth  of  the  point  of 

2  9 
entrance  below  the  level  of  the  head  race.     The  direction  of 

the  velocity  v  will  be  obtained  by  drawing  through  the  point 
of  entrance,  which  we  have  just  found  (since  it  is  on  the  ex- 


BREAST   WHEELS.  43 

terior  circumference  of  the  wheel,  and  on  a  known  horizontal 
line),  a  line  making  an  angle  of  30°  with  the  said  circumfer- 
ence. The  threads  should  then  be  obliged  to  take  this  direc- 
tion by  means  of  a  canal  of  from  Om.50  to  Om.60  in  length,  the 
last  element  of  which  should  be  tangent  to  v,  and  into  which 
the  water  would  flow,  either  by  passing  under  a  sluice,  or  by 
flowing  out  without  any  obstacle,  the  bottom  of  this  canal 
being  then  the  sole  of  the  weir. 

(c.)  Position  of  the  flume  as  regards  the  head  race  and  tail 
race. — We  have  only  to  repeat  here  what  has  already  been  said 
in  discussing  the  undershot  wheel  with  plane  floats.  When 
the  water  possesses,  on  leaving  the  wheel,  a  sensible  velocity, 
it  is  well  to  place  the  level  of  the  portion  between  the  floats, 
which  is  directly  under  the  axle,  below  the  level  of  the  lower 

2  u* 
section,  by  a  quantity  equal  to  -  —  ,u  being  the  velocity  of 

the  wheel  at  the  circumference.  The  velocity  u,  having  a 
value  previously  determined,  as  well  as  the  height  h  of  the 
water  between  the  floats,  the  situation  of  the  bottom  of  the 
flume  is  then  also  fixed,  at  least  that  part  below  the  axle.  On 
the  up-stream  side  this  bottom  has  a  circular  profile,  with  a 
radius  sensibly  equal  to  that  of  the  wheel ;  it  should  be  joined 
with  the  tail  race  in  accordance  with  M.  Belanger's  rules,  of 
which  we  have  previously  spoken  (No.  4). 

When  the  wheel  is  moving  slowly,  u  being  about  lm.30, 

-  - —  gives  a  depth  a  little  below  Om.06.  However  great  the 
6  2,  g 

head  may  be,  the  loss  or  gain  of  Om.06  is  of  but  small  import- 
ance, and  we  would  then  be  able,  by  economy,  to  do  away  with 
the  portion  of  the  race  beyond  the  wheel,  or  at  least  to  do 
without  a  masonry-lined  canal  with  a  regular  section.  The 
level  of  the  water  between  the  floats  would  then  coincide  with 


44  BREAST    WHEELS. 

that  of  the  tail  race.  But  in  the  case  of  large  values  of  u,  es- 
pecially if,  at  the  same  time,  we  have  only  a  slight  head,  the 

2  u* 

height may  afford  a  sensible  gain  that  we  should  not 

3  2  g 

neglect. 

(d.)  To  calculate  the  dynamic  effect  of  a  head  of  water 
which  sets  a  breast  wheel  in  motion. — Calling  P  the  weight  of  the 
water  expended  in  a  second,  H  the  head  measured  between  the 
two  portions  of  the  canal  supposed  to  be  in  a  state  of  rest,  we 
know  (No.  2)  that  the  dynamic  effect  Te  of  the  head,  during 
each  second,  has  for  its  value  the  product  of  the  weight  P  by 
the  height  H,  diminished  by  the  mean  loss  of  head  that  the, 
liquid  molecules  sustain  by  reason  of  friction  in  their  passage 
from  one  portion  of  the  canal  to  the  other.  Moreover,  this  loss 
of  head  is  composed  of  several  portions,  which  we  shall  now 
proceed  to  analyze,  using  the  notation  adopted  (No.  7,  &, 

*,  <•)• 

1st.  Loss  of  head  between  the  head  race  and  the  point  at 
which  the  water  enters  the  wheel. — This  is  reduced  by  avoiding, 
by  means  of  rounded  outlines,  contractions  followed  by  a  sud- 
den expansion,  and  by  placing  the  point  of  entrance  as  near 
the  head  race  as  possible.  Nevertheless  there  is  always  a  cer- 
tain loss :  this  we  may  assume  at  first  sight  as  comprised 

v*  v2 

between  0.05  - —  and  0.1—-  .     If  there  were  a  canal  between 
2  g  2  g 

the  wheel  and  the  head  race,  we  should  have  to  take  the  fric- 
tion in  this  canal  into  consideration,  by  a  calculation  analogous 
to  that  which  will  be  presently  given  for  the  circular  flume. 

2d.  Loss  of  head  due  to  the  introduction  of  the  water. — After 
what  has  been  said  (in  No.  3),  this  loss  may  be  valued  at 

- — ,  or  at  - —  (va  +  u*  —  2  u  v  cos  7).  In  the  case  of  slowly 
2  g  2  g 


BREAST   WHEELS.  45 


moving  wheels  it  is  of  slight  importance,  like  all  the  other 

losses  that  we  consider;    in  one  moving  quickly,  if  y  '=^Q^  -^^^ 

v  =    -  ,  we  shall  have 
cos/ 


_  _    tan,  = 


Library. 


When  the  precaution  is  taken  to  introduce  the  watcattfrjth  a 
relative  velocity  tangent  to  the  floats,  and  a  surface  is  presented 
along  which  it  can  ascend  in  virtue  of  this  relative  velocity,  it 
is  probable  that  the  action  of  gravity  assists  in  overcoming  this 
velocity,  and  thus  so  much  less  effect  will  be  consumed  in  a 
violent  disturbance  of  the  water.  This  consideration  justifies 
the  use  of  polygonal  floats,  such  as  are  shown  in  (Fig.  6),  the 
arrangement  of  which  was  contrived  by  M.  Belanger  in  1819. 
They  are  composed  of  three  planes  making  angles  of  45  degrees 
with  each  other,  of  which  the  furthest  from  the  centre  is  in  the 
direction  of  a  radius,  the  nearest  touches  the  circumference  of 
the  ring,  and  the  third  connects  the  two  others.  The  planes 
that  are  fixed  to  the  crown  have  vacant  spaces  left  between 
them  to  facilitate  the  disengagement  of  the  air.  The  point  of 
entrance  of  the  water  must,  moreover,  be  lower  than  the  centre 
of  the  wheel,  in  order  that  the  water  introduced  may  be 
received  upon  an  ascending  inclined  plane. 

Data  are  wanting  to  estimate  the  effect  produced  by  the  use 
of  this  kind  of  float. 

3d.  Loss  of  head  produced  by  the  friction  of  the  water 
against  the  circular  flume.  —  We  know  that  the  friction  of  a 
current  on  its  bed,  per  square  yard,  is  expressed  by  0.4  Ua,  U 
being  the  mean  velocity.  Moreover,  calling  Y  and  W  the 
velocities  at  the  surface  and  at  the  bottom,  it  has  been  found 
that  these  quantities  are  connected  by  the  approximate  rela- 
tions : 


46  BREAST    WHEELS. 

U  =  -(V  +  W),  U  =  0.80  Y; 
2 

whence 

The  friction  per  square  yard  of  bed  would  then  be 
0.4.  ^  Wa  or  0.71  Wa. 

Now,  if  L  be  the  length  of  the  circular  race,  L  (b  4-  2  h)  will 
be  the  surface  wet,  and  the  entire  friction  is  expressed  by  0.71 
L  (b  4-  2  h)  u*,  since  the  velocity  at  the  bottom  is  the  same  as 
the  velocity  at  the  circumference  of  the  wheel.  All  the  points 
of  application  of  the  forces  that  compose  this  friction  moving 
with  the  velocity  u,  their  negative  work  in  the  unit  of  time 
will  be  0.71  L  (b  -f-  2  h)  u3 ;  we  shall  obtain  the  corresponding 
loss  of  head  by  dividing  by  P,  or  by  1000  bhu,  which  gives 

0.00071  ^±4^-,  or  else  0.014  M*±  lAl^L. 
b  h  b  h         2  g 

4th.  Loss  of  head  from  the  point  just  beneath  the  axle  to  the 
tail  race. — If  the  level  of  the  water  between  the  floats,  beneath 
the  axle,  exceeds  by  a  height  v\  the  level  of  the  portion  lower 
down,  the  velocity  of  the  water  will  increase  on  account  of  this 
difference  of  level.  This  velocity,  sensibly  equal  to  u  on  leav- 
ing the  wheel,  will  become  Vu*  +  2  g  *i,  for  the  point  at  which 
the  water  begins  to  enter  the  lower  portion.  As  we  have  seen 

(No.  3),  a  loss  of  head  *)  +  - —  corresponds  to  this  velocity. 

2i  o 

When  we  adopt  the  arrangement  of  the  tail  race  recom- 
mended by  M.  Belanger  (No.  4),  i\  becomes  negative  and  equal 

to  -  - — ;  the  loss  then  reduces  to  -  — -. 
o  A g  32^ 

(e)   Some  practical  data. — With  regard  to  the  number  of 


BREAST    WHEELS.  47 

arms,  the  same  rule  here  applies  as  to  undershot  wheels 
(No.  4). 

The  number  of  floats  is  a  multiple  of  the  number  of  arms ; 
their  distance  apart  may  be  from  once  and  a  third  to  once  and 
a  half  the  head  above  the  top  of  the  weir,  in  the  case  of  slowly 
moving  wheels.  In  wheels  of  rapid  motion,  it  will  always  be 
necessary  to  take  this  distance  apart  a  little  greater  at  the  por- 
tion of  the  circumference  intercepted  by  the  stratum  of  water 
that  falls  upon  the  wheel.  It  is  not  well  to  place  the  floats  too 
far  apart,  because  some  of  the  threads  might  fall  from  quite  a 
considerable  height  before  reaching  them ;  it  is  also  bad  to 
place  them  too  close  together,  because  the  water  could  with 
difficulty  enter  the  wheel,  and  a  portion  would  be  thrown  off. 

The  depth  of  the  floats  in  the  direction  of  the  radius  is  but 
little  greater  than  Om.TO.  In  its  normal  condition,  the  interior 
capacity  formed  by  any  two  consecutive  floats  should  be  very 
nearly  double  the  volume  of  water  that  they  contain  :  we 
might  then  take  the  depth  in  question  equal  to  2  A,  whenever 
h  is  not  greater  than  Om.35. 

The  diameter  of  the  wheel  should  be  at  least  3m.50  ;  it  is  sel- 
dom greater  than  6  or  7  metres.  The  axis  of  the  arbor  is 
placed  a  little  above  the  level  of  the  up-stream  portion  of  the 
canal. 

Breast  wheels  are  suitable  for  falls  of  from  1  to  2  metres,  or 
even  2m.50.  Beyond  these  limits  they  can  still  be  frequently 
used  to  advantage. 

When  a  breast  wheel  of  slow  motion  is  well  organized,  the 
dynamic  effect  of  the  head  may  approximate  quite  near  to  what 
is  due  to  the  entire  head.  General  Morin,  in  some  experiments 
on  these  wheels,  found  an  effective  delivery  of  0.93.  But, 
since  the  measure  of  the  expenditure  of  water  is  always  some- 
what involved  in  uncertainty,  and  consequently  the  total  effect 


48  BREAST    WHEELS. 

due  to  a  given  head  can  be  but  imperfectly  ascertained,  it  will 
be  prudent  not  to  count  beforehand  on  an  effective  delivery,  in 
practice,  greater  than  0.80. 

8.  Example  of  calculations  for  a  rapidly  moving  Breast 
wheel. — The  wheel  in  question  was  experimented  on  by  Gene- 
ral Morin  ;  it  belonged  to  the  foundry  at  Toulouse. 

The  water  left  the  head  race  under  a  sluice  that  was  raised 
(y*.14:7  above  the  sole,  which  was  lm.423  below  the  above-men- 
tioned level.  The  orifice  being  prolonged  by  a  very  nearly 
horizontal  race,  the  velocity  v  with  which  the  water  reaches  the 
wheel  will  be  due  to  the  head  at  the  upper  portion  of  the  ori- 
fice, except  a  co -efficient  of  correction  very  nearly  1,  which  we 
will  value  at  0.95  ;  hence 


v  =  0.95  V  2g  (lm.423  -  Om.147)  =  4m.75. 
The  velocity  at  the  circumference  of  the  wheel  was  u  =  3m.06, 
and  the  angle  made  by  u  and  -y,  at  the  point  of  introduction, 
was  valued  at  30°.     It  follows  directly  from  this,  that  the  loss 
of  head  sustained  in  bringing  the  water  from  the  head  race  to 

the  wheel  should  be  i  of  the  head  (lm.423  —  Om.147)  or  Om.13  ; 

we  can  also  calculate  that  produced  by  the  introduction  of  the 
water,  which  was  expressed  by  (No.  7,  d) 

~  =  ~  (V  +  u?  -  2  uv  cos  300>)  =  Om.34. 
2  g        2  g  \  ) 

The  level  of  the  water  between  the  floats,  just  below  the 
axle,  being  at  the  level  in  the  down-stream  portion  of  the  canal, 

u* 
we  must  again  count  upon  a  loss  equal  to  - —  (No.   7,   <f),  or 

2  9 
Om.48. 

Finally,  there  is  a  loss  in  the  circular  flume.  The  depth  of 
the  water  beneath  the  axle  and  the  breadth  of  the  wheel  being 
valued  respectively  at  Om.20  and  lm.55,  and  the  length  of  the 


BKEAST    WHEELS.  4-9 

flume  being  2m.50,  we  find  for  this  loss  0.014.  2'5°  XJ±^  ^_ 

0.20  x  1.55  2  g> 

or  Om.ll. 

All  these  losses  together  make  up  a  head  of  Om.13  -f  Om.34 
+  Om.48  +  Om.ll,  or  of  lm.06.  The  head  being  lm.72,  we  see 
that  the  productive  force  as  calculated  would  be  only 

•^  Y2 1  06 

— — :r-=^ — ,  that  is,  0.38  ;  M.  Morin  found  experimentally  0.41, 
1.7.4 

a  number  which  corresponds  to  an  available  head  of 

lm.T2  x  0.41  =  Om.705, 

instead  of  Om.66,  which  the  preceding  calculation  gives.  This 
difference,  otherwise  hardly  noticeable,  from  Om.045,  belongs 
probably  to  a  somewhat  greater  value  of  the  head  lost  by  the 
introduction  of  the  water ;  in  fact  we  have  seen  that,  in  certain 
cases,  a  portion  of  the  relative  velocity  could  be  annulled  by 
the  action  of  gravity,  which  would  diminish  by  so  much  the 
disturbance  within  the  floats,  and  would  give  rise  to  a  smaller 
loss  of  head. 

To  increase  the  effective  delivery  of  this  wheel  without 
changing  its  velocity,  the  following  arrangements  might  have 

been  made :  First,  to  place  the  flume  at  —  — ,  or   Om.32  lower 

6  2g 

than  its  actual  position,  taking  care  to  arrange  the  race  as 
described  by  M.  Belanger  (No.  4) — that  is,  without  any  sudden 
variation  in  section,  and  with  a  bottom  having  a  moderate 
slope,  to  its  junction  with  the  tail  race ;  then  to  raise  the  point 
of  entrance  of  the  water  so  as  to  reduce  the  velocity  v  to 

u     .  or  8m'°6  =  3m.52.     The  loss  from  the  wheel  to  the 


cos  30°       0.866 

tail  race  would  then  have  been  reduced  to  -  - —  or  to  Om.16, 

32^ 

instead  of  Om.48  ;  the  loss  for  the  entrance  of  the  water  would 


50  BREAST   WHEELS. 

also  be  reduced  to  the  same  value  Om.16  instead  of  Om.34,  which 
would  procure  a  total  benefit  of  Om.50.  The  other  losses  remain- 
ing sensibly  the  same,  the  available  head  would  be  Om.66  +  Om.50 
=  lm.16,  and  the  effective  delivery  would  be  raised  to  about 


1.72 


OVERSHOT  WHEELS. 


9.  Wheels  with  buckets,  or  over-shot  wheels. — These  wheels 
are  not,  like  the  preceding,  set  in  a  canal.  The  water  is  let  in 
at  the  upper  portion ;  it  enters  the  buckets,  which  are,  as  it  were, 
basins  formed  by  two  consecutive  floats,  terminated  at  the  sides 
by  the  annular  rings,  and  closed  at  the  bottom  or  sole  by  a  con- 
tinuous cylindrical  surface  concentric  with  the  wheel.  The 
questions  that  the  organization  of  this  kind  of  motor  present 
are  as  follows : 


FIG.  8. 


(a)  Introduction  of  water  into  the  wheel. — Two  arrange- 
ments are  employed  which  are  represented  hereafter  (Figs.  8, 
9).  In  (Fig.  8)  the  top  D  of  the  wheel  is  placed  a  little  below 


52  OVERSHOT   WHEELS. 

the  level  K  N  of  the  pond,  at  Oft.60  to  Oft.75  lower  than  that 
level ;  the  water  is  led  to  a  point  C,  situated  about  lft.50  up- 
stream, and  is  delivered  directly  above  the  axle,  by  means  of  a 
canal  A  B,  or  pen  trough  made  of  planks,  terminated  by  a  very 
thin  metallic  plate  B  C,  which,  being  prolonged,  would  be  very 
nearly  tangent  at  D  to  the  exterior  circumference.  The  lateral 
boundaries  of  the  canal  ABC  are  prolonged  1  metre  beyond 
the  point  C,  to  prevent  the  water  from  falling  outside  of  the 
wheel.  The  water  passes  over  the  distance  C  D  in  virtue  of  its 
acquired  velocity,  and  enters  the  wheel  nearly  at  the  top.  As 
quite  a  narrow  opening  only  is  left  between  the  soles  of  the 
buckets,  the  water  that  flows  in  the  canal  A  B  C  is  given  the 
form  of  a  thin  stratum,  by  making  it  pass  under  a  sluice  placed 
near  A,  and  which  is  raised  only  about  Om.06  or  Om.10.  This 
sluice  presents  an  orifice  with  rounded  edges,  so  as  to  avoid  the 
eddies  consequent  to  the  exit  of  the  liquid  threads.  As  has 
been  shown,  there  is  little  difference  in  height  between  the 
point  at  which  the  water  enters  and  the  level  of  the  head  race ; 
consequently  the  water  enters  the  wheel  with  a  slight  absolute 
velocity,  and  if  the  wheel  turn  slowly,  as  it  should  do,  to  attain 
a  good  effective  delivery,  the  relative  velocity  will  itself  be 
slight,  as  well  as  the  loss  of  work  that  it  involves. 

The  arrangement  in  (Fig.  9),  which  has  been  frequently  em- 
ployed, does  not  appear  to  be  so  good ;  but  we  are  sometimes 
obliged  to  make  use  of  it  if  the  pond  level  is  very  variable. 
This  portion  is  terminated  near  the  wheel  by  a  wooden  shutter 
A  B,  with  openings  C  C,  having  vertical  faces  like  those  of  a 
window-blind;  a  movable  sluice  allows  of  covering, as  many 
of  these  openings  as  may  be  requisite,  so  as  to  expend  only 
the  disposable  volume  of  water.  The  inconvenience  of  this 
method  is,  that  the  water  falls  through  sufficient  height  into 
the  buckets  to  give  it  a  considerable  increase  of  velocity ;  the 


OVERSHOT   WHEELS. 


53 


disturbance  of  the  water  in  the  wheel  thus  becomes  much 
greater.  It  tends,  moreover,  for  the  same  head,  to  increase  the 
diameter  of  the  wheel,  which  makes  it  more  heavy  and  expen- 


FIG.  9. 

sive.  Besides,  the  point  at  which  the  buckets  take  a  sufficient 
inclination  to  begin  to  discharge  the  water  in  them  is  situated 
at  a  greater  height  above  the  lowest  point  of  the  wheel,  because 
this  height  is  proportional  to  the  diameter ;  there  is  thus,  then, 
a  greater  loss  of  head,  seeing  that  the  work  of  the  weight  of  the 
molecules  that  have  left  the  buckets,  whilst  they  are  falling 
into  the  race  below,  is  evidently  lost  to  the  wheel. 

(5)  Shape  of  the  surface  of  the  water  in  the  buckets  ;  velo- 
city of  the  wheel. — It  can  be  shown  that  a  heavy  homogeneous 
liquid  cannot  be  in  equilibrio  relatively  to  a  system  that  turns 
uniformly  about  a  horizontal  axis.  If,  however,  we  admit  that 
the  relative  equilibrium  of  the  water  can  exist  approximately  in 
the  buckets,  which  may  arise  when  the  disturbance  due  to  the 
entrance  of  the  liquid  has  nearly  ceased,  we  can  determine  the 
shape  assumed  by  the  free  surface  as  follows  : 


OVERSHOT    WHEELS. 


Let  M  (Fig.  10)  be  a  liquid  molecule,  having  a  mass  ra,  situ- 
ated at  the  distance  O  M  =  r  from  the  axis  of  rotation  O  ;  it  is 
in  equilibrio  relatively  with  a  system  which  turns  around  this 
axis  with  an  angular  velocity  u.  This  equilibrium  exists  under 
the  action  :  1st,  of  the  weight  m  g,  which  acts  vertically  along 
the  line  M  G ;  2d,  of  the  centrifugal  force  m  wa  r,  along  the 
prolongation  M  C  of  O  M,  an  apparent  force  to  be  introduced, 
as  regards  solely  a  relative  equilibrium ;  3d,  of  the  pressures 
produced  by  the  surrounding  molecules.  We  know  from  the 
principles  of  hydrostatics  that  the  resultant  of  the  two  tirst 
forces  is  normal  to  the  surface  level  (or  of  equal  pressure)  which 

passes  through  M.  If,  then,  M 
be  found  at  the  free  surface,  as 
the  pressure  there  is  entirely 
the  atmospheric  pressure,  the 
resultant  in  question  will  be 
normal  to  this  surface.  Let  us 
take  M  G  =  m  g,  M  C  =  m  u V, 
the  diagonal  M  B  of  the  paral- 
lelogram M'G  B  C  represent  the 
resultant  of  m  g  and  of  m  wa  r, 
and  consequently  it  is  normal  to 
the  free  surface.  Moreover,  pro- 
longing the  vertical  O  A  until  it  intersects  this  normal,  we 
obtain  from  the  property  of  similar  triangles, 


^K 

!  V 


Fie.  10. 


OA 
"MG 


OM 
GB' 


whence 


n  A       MG  x  OM        mgr  __  g_, 

^J    -£X      —      f^      -r~i  a  —  O     J 

G  B  —  -  •  M      •  • 


m  u  r 

hence  the  distance  O  A  is  constant,  which  shows  that  in  a  plane 
section  perpendicular  to  the  axis  all  the  normals  to  the  free 


OVEESHOT   WHEELS.  55 

surface  meet  at  the  same  point.  The  profile  of  the  free  sur- 
face, if  there  be  relative  equilibrium,  is  then  necessarily  a  circle 
described  from  the  point  A  as  a  centre. 

This  result  proves  that,  in  effect,  the  relative  equilibrium  is, 
strictly  speaking,  impossible ;  for,  in  proportion  as  the  bucket 
leaves  its  place,  the  point  A  not  changing  position,  the  free 
surface  would  have  an  increasing  radius,  which  is  incompatible 
with  the  hypothesis  of  relative  equilibrium,  since  in  this  case 
the  form  of  the  free  surface  ought  not  to  change.  The  form 
that  has  been  determined  is  that  which  the  water  endeavors  to 
assume  without  being  able  to  preserve  it. 

To  finish  determining  the  circle  which  limits  the  water  in  a' 
given  bucket,  a  circle  of  which  we  as  yet  know  only  the  centre 
A,  we  must  take  into  consideration  the  quantity  of  water  that 
the  bucket  is  to  hold.  To  this  end,  let  N  be  the  number  of 
buckets  filled,  &  the  breadth  of  the  wheel  parallel  to  the  axis, 
Q  the  volume  expended  by  the  pond  per  second.  Each  bucket 

2  if 
occupies  on  the  circumference  an  angle  -^=-  (expressed  in  terms 

of  an  arc  of  a  circle  having  a  radius  1),  and  as  the  wheel  turns 
with  an  angular  velocity  w,  — —  will  represent  the  number  of 

buckets  filled  in  a  unit  of  time.     Each  bucket  then  contains  a 

2  tf  Q 
volume  — «£,  so  that  the  area  occupied  by  the  water  in  the 

2  *  Q 

cross  section  of  the  bucket  has  for  its  value  -I — ^r.     The  arc 

b  w  .N 

D  M  E  will  then  be  determined,  since  we  know  its  centre  and 
the  surface  D  M  E  F  1  which  it  must  intercept  in  the  given 
profile  of  the  bucket. 

In  a  certain  position  V  E'  F'  of  the  bucket,  the  free  surface, 
determined  in  the  way  just  mentioned,  just  touches  the  edge  of 


56  OVERSHOT   WHEELS. 

the  exterior  side  E' ;  this  position  may  be  found  by  trial.  As 
soon  as  the  bucket  passes  it,  the  water  begins  to  run  out ;  for 
every  position  below  this,  it  is  clear  that  the  free  surface  will 
have  for  a  profile  a  circle  having  A  for  a  centre  and  just  touch- 
ing the  outer  edge  of  the  bucket,  and  which  will  allow  the 
volume  of  water  remaining  in  the  bucket  to  be  determined. 
When  the  circle  in  question  passes  entirely  below  the  profile  of 
the  bucket,  the  discharge  will  be  complete. 

In  practice,  if  the  question  relates  to  wheels  possessing  only 

a  slow  angular  velocity,  -^-  will  be  so  great  that  the  circles  de- 

OJ 

scribed  from  A  as  a  centre,  to  limit  the  surface  of  the  water  in 
the  buckets,  may  be  assumed  as  horizontal  lines.  Example : 
The  wheel  being  four  metres  in  diameter,  and  having  a  velocity 

of  1  metre  at  the  circumference,  then  a  —  -  and  -^  =  39m.24, 
or  the  distance  of  the  centre  A  above  the  axis. 

When  an  overshot  wheel  turns  rapidly,  the  distance  —^  may 

W 

become  so  small  that  the  free  surface  may  present  a  noticeable 
concavity  below  the  horizontal ;  thus  the  more  the  angular 
velocity  increases  the  less  water  the  bucket  can  hold  in  a  given 
position,  which  is  easily  seen,  since  the  centrifugal  force  becomes 
greater  and  greater,  and  this  force  tends  to  throw  the  water  out 
of  the  bucket.  This  is  an  inconvenience  attendant  upon  wheels 
that  turn  rapidly ;  they  lose  a  great  deal  of  water  by  spilling, 
and  consequently  yield  a  smaller  effective  delivery. 

The  losses  of  head  produced  by  the  introduction  of  the  water 
into  the  buckets,  and  by  the  velocity  of  the  water  when  it 
leaves  the  wheel,  also  increase  with  the  angular  velocity.  We 
will  then  be  led,  in  order  to  economize  the  motive  force  as 
much  as  possible,  to  make  the  wheel  turn  very  slowly.  But 


OVERSHOT    WHEELS.  57 

we  have  already  seen,  in  speaking  of  breast  wheels,  that  it  is 
not  well  to  make  a  water  wheel  move  very  slow,  because,  in 
order  to  use  up  an  appreciable  volume  of  the  water,  it  would 
be  necessary  to  establish  a  machine  of  immense  size.  A  velo- 
city of  from  1  metre  to  lm.50  at  the  circumference  gives  good 
results. 

(c)  Breadth  of  the  wheel ;  depth  of  the  buckets  in  the  direc- 
tion of  the  radius. — We  have  said  above  that,  in  a  well- 
arranged  wheel,  the  water  leaves  the  up-stream  portion  of  the 
troughs  by  passing  under  a  sluice  raised  from  Om.06  to  Om.10 
above  the  sole ;  which  is  itself  from  Om.20  to  Om.25  below  the 
level  of  this  portion.  If  we  call 

Q  ;he  expenditure  per  second ; 

I  the  breadth  of  the  wheel,  and  of  the  orifice  under  the 
sluice ; 

x  the  height  to  which  the  sluice  gate  is  raised ; 

h  the  depth  of  the  sole  below  the  level  of  the  water  in  the 
up-stream  section ; 

v  the  velocity  with  which  the  water  leaves  the  sluice ; 
the  velocity  v  will  be  due  very  nearly  to  the  head  h  —  x;  and  as 
the  adjustments  are  so  arranged  as  to  have  but  little  contrac- 
tion, we  can  place 

Q  =  0.95  Ix  V  2  g  (h  —  x\ 

the  co-efficient  0.95  being  intended  to  account,  at  a  rough  esti- 
mate, for  the  loss  of  head  that  water  always  undergoes  in  any 
movement  whatever,  and  for  the  contraction  that  would  yet 
partially  exist.  By  making,  in  this  expression,  h  =  Om.20,  x  = 

Om.06,  we  deduce  ~  =  Om.095  ;  in  like  manner,  for  h  =  Om.25, 
o 

x  —  Om.10,  we  find  ~  =  Om.163 ;  that  is,  with  the  sluice  ar- 
o 

ranged  as  we  have  have  said,  we  can  expend  from  95  to  163 


58  OVERSHOT  WHEELS. 

litres  per  metre  of  breadth  of  the  wheel.  It  would  be  easy  to 
expend  less  than  95  litres,  by  diminishing  A  and  x  a  little  ;  we 
can,  when  necessary,  expend  more  than  163  litres  by  inverse 
means.  But  experience  shows  that,  to  be  in  the  best  condition, 
the  expenditure  should  be  but  little  more  than  100  litres  per 
metre  of  breadth ;  for  otherwise  we  might  be  led  either  to  make 
deep  buckets,  or  to  cause  the  wheel  to  turn  rapidly,  which 
would  tend  to  increase  the  velocity  of  the  water  when  it  enters 
or  leaves  the  wheel,  and  consequently  to  diminish  the  effectire 
delivery. 

To  show  the  relation  that  exists  between  the  depth  p  of  the 

buckets  in  the  direction  of  the  radius,  and  the  expense  -j-  per 

o 

yard  in  breadth,  let  us  preserve  the  notation  already  employed 
in  the  present  number,  and  furthermore  let  us  call 

R  the  radius  of  the  wheel ; 

u  its  velocity  at  the  circumference ; 

C  — — =^r—  the  distance  of  the  buckets  apart ;  c  their  thick- 
ness. 

The  volume  of  a  bucket  will  be  equal  to  the  product  of  its 
three  mean  dimensions,  viz. :  its  length  £,  its  depth />,  and  its 

breadth  C  (~L  —  ^-U)  —  c  ;  this  volume  has  then  for  its  value, 

p  b  C  fl  —  -^~=  — -  ).     Moreover  it  would  be  well,  to  retard 

the  discharge  from  the  bucket,  not  to  have  it  more  than  one- 
third  full ;  the  volume  of  the  water  it  contains  would  then  be 

-p  6  C  (l  — -~  —  —  Y  and,  as  we  have  previously  seen,  by 

— _r_-,  there  obtains 

«  N 


OVERSHOT  WHEELS.  59 


,  ,  ~       2  *  R      ,          -w 

whence,  because  C  =  ——  and  w  =       ; 


As  the  factor  in  the  parenthesis  in  the  second  member  differs 
but  little  from  1,  we  may  simply  place 

Q      1 

T  =  ^u' 

This  equation  shows  that  when  -^  is  large,  one  of  the  factors  p 

or  u  must  be  so  too.     For  example,  if  ~  =  0.100  litres,  and  u 

—  1  metre,  we  find  p  =  Om.30.  It  is  desirable  that  p  should 
not  much  exceed  Om.30.  However,  if  we  had  an  ample  supply 
of  water  to  expend,  we  might  either  go  beyond  this  limit,  or 
use  a  faster  wheel,  or  finally  fill  the  buckets  more  than  one- 
third. 

The  expenditure  per  metre  of  breadth  having  been  fixed, 
from  what  precedes,  as  much  as  possible  below  100  litres  per 
second,  the  breadth  of  the  wheel  results  naturally  from  the 
total  volume  of  water  to  be  expended.  It  is  seldom  that 
wheels  having  a  greater  breadth  than  5  metres  are  con- 
structed. 

(d)  Geometrical  outline  of  ike  "buckets. — The  distance  of  the 
buckets  apart  is  a  little  greater  than  their  depth ;  generally, 
this  last  dimension  is  from  Om.25  to  Om.28,  and  the  other  about 
from  Om.32  to  Om.35.  Their  number  must  be  a  multiple  of  the 
number  of  arms  for  facilitating  the  connections,  unless  the 
crown  and  arms  are  composed  of  a  single  piece. 

As  to  their  profile,  the  annexed  outline  is  frequently  made 


60  OVERSHOT    WHEELS. 

use  of  (Fig.  11).  After  dividing  the  exterior  circumference 
O  A  into  portions  A  A7,  A'  A",  all  equal  to  the  distance  of 
the  buckets  apart,  we  take  A  D  =jp,  the  depth  of  the  buckets 
in  the  direction  of  the  radius,  and  describe  the  circumference 
O  D ;  a  third  circumference  is  then  drawn,  O  B,  at  equal  dis- 
tances from  the  first  two.  The  radii  O  A,  O  A',  O  A"  .  .  . 


W 
o 

FIG.  ll. 


being  then  drawn  through  the  points  of  division,  A  B',  A'  B", 
.  .  .  will  be  joined,  and  we  shall  then  have  the  profiles  A  B'  D', 
A'  B"  D",  .  .  .  which,  excepting  the  thickness,  will  be  those 
of  the  buckets. 

Skilful  constructors  think  that,  instead  of  the  lines  such  as 
A  B',  A'  B",  ...  we  might  employ  the  lines  a  B',  a!  B",  .  . 
which  produce  a  certain  degree  of  mutual  covering  between 
the  buckets  ;  in  like  manner  for  the  lines  B  D,  B'  D',  B"  D", 
.  .  .  ,  the  inclined  right  lines  B  d,  B'  df,  ~B"  d" ,  .  .  .  ,  have 
been  sometimes  substituted.  These  two  changes  have  the  one 
end,  that  of  increasing  the  depth  of  the  buckets  in  the  direction 
parallel  to  the  circumference,  and  consequently  to  retard  the 
emptying.  They  are  inconvenient,  because  they  make  the 
construction  more  difficult ;  besides,  this  overlapping  A  a  must 


OVERSHOT   WHEELS.  61 

not  be  carried  to  excess,  otherwise  the  remaining  free  space 
between  the  point  B  and  the  side  a  B'  would  perhaps  be  too 
much  diminished.  This  minimum  distance  should  be  a  few 
centimetres  greater  than  the  height  to  which  the  sluice  gate  is 
raised,  in  order  that  the  water  may  enter  well  into  the  wheel, 
and  not  be  thrown  to  the  outside. 

When  the  buckets  are  made  of  sheet-iron,  the  broken  pro- 
files just  mentioned  are  replaced  by  curved  profiles,  which 
should  differ  as  little  as  may  be  from  them. 

Wooden  buckets  are  generally  from  15  to  30  millimetres  in 
thickness ;  the  sheet-iron  ones  are  only  from  2  to  4  millimetres, 
which  increases  slightly  their  capacity,  all  other  things  being 
equal.  They  are  limited  at  the  sides  by  the  annular  rings, 
which  are  fastened  to  the  axle  by  arms,  which  increase  in  num- 
ber with  the  diameter  of  the  wheel.  They  present  a  continu- 
ous bottom  or  sole  throughout  the  entire  circumference 
D  D'  D".  .  .  .  ;  in  very  large  wheels  this  bottom  must  be 
sustained  by  supports  at  one  or  two  points  placed  between  the 
exterior  crowns.  We  might  also  use  in  this  case  one  or  two 
intermediate  crowns. 

(e)  To  calculate  the  dynamic  effect  of  a  head  that  causes  an 
overshot  ivheel  to  turn. — The  two  main  causes  which  give  rise 
to  the  losses  of  head  to  be  subtracted  from  the  entire  head,  to 
obtain  the  head  that  is  turned  to  account,  are  the  relative  velo- 
city when  the  water  enters  the  wheel,  and  that  which  it  pos- 
sesses at  the  moment  that  it  falls  to  the  level  of  the  tail  race. 

It  is  almost  impossible  to  obtain  an  accurate  value  of  the 
first.  During  the  time  that  a  bucket  is  being  filled,  the  point 
of  entrance  of  the  molecules,  which  come  in  successively,  is 
changing  in  a  continuous  manner.  The  first  impinge  against 
the  solid  sides ;  those  that  come  after,  against  those  that  are 
already  in;  and  thence  result  phenomena  very  difficult  to 


62  OVERSHOT   WHEELS. 

analyze.     The  study  is  greatly  simplified  by  admitting,  as  we 

w* 
did  in  (No.  3),  that  the  height  —  —  ,  due  to  the  relative  velocity 

2  ff 

w  of  the  water  at  its  point  of  entrance,  represents  the  loss  of 
head  in  question.  Besides,  if  we  call  v  the  absolute  velocity 
of  the  water,  u  the  velocity  of  the  wheel,  7  the  angle  formed 
by  the  two  velocities  ;  as  w  is  the  third  side  of  the  triangle 
formed  by  v  and  u,  we  shall  have 

W*  =  U*  4-  v9  —  2  u  v  cos  7. 

In  reality,  the  impinging  of  the  water  on  the  wheel  takes  place 
at  different  points  along  the  depth  of  the  bucket.  Recollecting 
now  that  the  radius  of  the  wheel  is  great  compared  with  the 
thickness  of  the  shrouding  of  the  buckets,  this  will  not  mate- 
rially affect  u;  but  to  determine  v  and  7,  it  would  be  well  per- 
haps to  suppose  the  point  of  entrance,  not  at  the  exterior  cir- 
cumference, but  at  the  middle  of  the  depth  of  the  buckets. 

Let  us  pass  to  the  second  loss.  Let  a  molecule  of  the  mass 
m  leave  the  wheel  at  a  height  z  above  the  tail  race.  This 
molecule,  having  only  an  insensible  relative  motion  in  the 
bucket,  possesses,  at  the  moment  that  it  leaves  it,  the  velocity 
u  of  the  wheel,  and  at  the  moment  it  reaches  the  level  of  the 
tail  race  it  has  a  velocity  vf  equal  to  V  u*  +  2  g  z.  Then  it 
gradually  loses  all  its  velocity  while  moving  in  this  portion, 
without  its  piezometric  level  changing  (for  we  suppose  the  free 
surface  horizontal  in  this  portion)  ;  it  undergoes  then  a  loss  of 
head  equal  to 


For  all  the  molecules  composing  the  weight  P  expended  in  a 
second,   there  will    be  a  mean   loss   expressed   by  p-  2  m  g 


OVERSHOT   WHEELS.  63 

°r  e^8e  ^  2~~  +  p  2  m  ^  s>  tne  sum  2  including  all 

y 

the  molecules.  This  will  be  the  second  height  to  be  subtracted 
from  the  total  height  of  the  head ;  it  is  composed  of  two  terms, 
of  which  the  first  is  at  once  given,  and  it  only  remains  to  be 

seen  how  we  can  calculate  the  term  —  ^mgz,  which  expresses 
the  special  effect  of  the  emptying  of  the  buckets. 

The  quantity  -^  2  m  g  z  is  nothing  more   than   the   mean 

height  comprised  between  the  point  of  exit  of  a  molecule  and 
the  level  of  the  tail  race ;  as  the  circumstances  of  all  the  buck- 
ets are  exactly  the  same,  it  is  evidently  sufficient  to  seek  this 
mean  for  the  molecules  contained  in  one  bucket.  To  this  end, 
we  will  first  determine,  as  stated  above  (£),  the  positions  of  the 
bucket  at  which  the  emptying  begins  and  ends,  and,  for  a  cer- 
tain number  of  intermediate  positions,  we  will  ascertain  the 
amount  of  water  that  remains  in  the  bucket.  Let  then 

G  be  the  height  of  the  outer  edge  of  the  bucket  above  the 
level  of  the  tail  race  when  the  emptying  begins,  and  let  the 
bucket,  still  full,  hold  the  volume  of  water  q0  • 

G'  the  analogous  height  when  the  emptying  has  just  ended ; 

y  the  distance  that  this  same  edge  has  descended  whilst  the 
volume  of  water  qti  was  being  reduced  to  q. 

During  an  infinitely  small  displacement  of  the  wheel,  to 
which  the  descent  d  y  corresponds,  an  infinitely  small  volume 

—  d  q  is  emptied  out,  which  falls  into  the  tail  race  from  a 
height  G  —  y ;   the  mean  height  of  the  outflow  will  then  be 

—  f^0  (c  —  y)d  q.     Now  integrating  by  parts  there  obtains 

20         0 


64:  OVERSHOT    WHEELS. 

and,  observing  that  y  —  c  —  c'  and  y  ==  o  correspond  to  the 
limit  q  =  o  and  q  =  q^ 

q0c+f°_c,qdy  =  q0c-f°~    °  qdy; 
the  mean   sought   is   then  -p 2mg z  =  c  -         f  qdy. 

/{*    — —    /* 
q  d  y  will  be 

effected  by  Simpson's  method,  for  want  of  a  strict  analysis,  since 
we  have  the  means  of  determining  the  value  of  q  correspond- 
ing to  a  given  value  of  y.  Were  we  satisfied  with  a  greater  or 
less  approximation,  but  generally  one  sufficient,  we  could, 

under  the  sign/*,  replace  the  variable  q  by  the  mean  -  q^  of  its 

2t 

extreme  values ;  we  should  then  find, 

_2  mgz  =  c-  -  (c  -  c')  =  -  (c  +  </)• 

To  the  losses  already  determined  we  must  still  add  one  for 
bringing  the  water  from  the  upper  portion  of  the  canal  to  the 
wheel.  As  we  have  seen  in  examining  other  wheels,  it  will  be 
very  slight  if  a  good  arrangement  be  adopted  ;  its  value  would 

•y8 

then  be  0.1  - — . 
2<T 

(f)  Practical  suggestions. —  The  liquid  molecules  taking, 
during  their  fall  which  follows  the  discharge,  a  velocity  sensi- 
bly vertical,  there  are  scarcely  any  means  for  turning  this  velo- 
city to  account  by  a  counter-slope,  and  consequently  the  down- 
stream level  should  just  graze  the  bottom  of  the  wheel.  For 
wheels  that  turn  rapidly,  it  would  perhaps  be  advantageous  to 
set  them  in  a  mill  race,  which  would  only  allow  the  water  to 
escape  at  the  lower  portion,  and  with  a  velocity  nearly  horizon- 
tal ;  we  should  take  care  to  furnish  each  bucket  with  a  valve, 


OVERSHOT   WHEELS.  65 

placed  near  the  bottom,  and  opening  from  without  inwards,  to 
allow  the  air  to  enter  when  the  water  runs  out.  It  would  then 
be  possible  to  diminish  greatly  the  loss  of  head  occasioned  by 
the  velocity  that  the  water  possesses  on  leaving  the  wheel ;  but, 
on  the  other  hand,  we  would  increase,  in  no  small  degree,  the 
expense  of  erecting  the  wheel,  and  very  likely  also  that  of 
keeping  it  in  repair. 

Overshot  wheels  answer  very  well  for  heads  from  4  to  6 
metres  ;  less  than  3  metres  the  breast-wheel  is  to  be  preferred. 
Besides,  as  their  diameter  is  nearly  equal  to  the  height  of  the 
water  fall,  their  employment  would  become  practically  impos- 
sible for  very  high  falls. 

Experience  shows  that  with  an  overshot  wheel,  well  set  up 
and  moving  slowly,  the  productive  force  of  the  head  of  water 
may  rise  as  high  as  0.80,  and  sometimes  even  more.  But,  in 
wheels  that  turn  rapidly,  it  sometimes  falls  as  low  as  0.40. 


TUB    WHEELS. 


§  III.     WATER  WHEELS  WITH  VERTICAL  AXLES. 

10.  Old-fashioned  spoon  or  tub  wheels.  —  The  paddles  of 
spoon-wheels  are  of  slightly  concave  form  in  the  direction  of 
their  length  and  breadth.  They  are  arranged  around  a  vertical 
axle,  and  receive  in  an  almost  horizontal  direction  the  shock  of 
a  fluid  vein  which,  leaving  a  reservoir  with  a  great  head,  is  led 
near  the  wheel  by  a  wooden  trough.  To  obtain  a  greater 
action,  the  water  is  made  to  strike  against  the  concave  side  of 
the  paddles. 

Let  us  call 

v  the  absolute  velocity,  supposed  to  be  horizontal  and  per- 
pendicular to  the  paddle  struck,  of  the  vein  which  strikes  the 
wheel  ; 

w  the  section  of  this  vein  ; 

u  the  velocity  of  the  paddles  at  the  point  at  which  they  re- 
ceive the  shock  ; 

n  weight  of  the  cubic  metre  of  water. 

The  relative  velocity  of  the  water  and  paddles  will  be  hori- 
zontal and  equal  tov  —  u  ;  then,  if  this  phenomenon  be  assimi- 
lated to  that  of  a  fluid  vein  impinging  against  a  plane,  the 

force  exerted  on  the   wheel  will  be  n  w  ^  —  —  ^-  ;  the  work 


which  this  force  performs  on  the  wheel  in  the  unit  of  time  will 


^—    —'- 


be  expressed  by  n  w  —  —    —'-,  a  quantity  sensibly  equal  to  the 


TUB    WHEELS. 


67 


dynamic  effect  of  the  head  (ISTo.  2).  This  expression  varies 
with  u,  and  reaches  its  maximum  for  u  =  -  v ;  this  maximum 

o 

is  n  u  —  — ,  or,  seeing  that  n  w  v  gives  the  weight  P  expended 

2i  i   (j 

per  second,  —  P  — .     The  height  - —  can  only  be  a  fraction 

of  the   head   H;  hence,  the  dynamic  effect   is   found  below 

8  8 

—  P  H,  and  the   effective  delivery  below  — -,  or  about  0.30. 

This  number,  moreover,  cannot  be  considered  as  strictly  cor- 
rect, because  the  imperfectness  of  the  theories  relating  to  the 
resistance  of  fluids  has  caused  us  to  give  a  more  or  less  approxi- 
mate value  for  the  reciprocal  action  of  the  water  and  the  wheel ; 
however,  experience  confirms  the  result  of  calculation,  at  least 
inasmuch  as  it  indicates  for  this  class  of  motors  an  effective 
delivery  that  is  always  very  small,  varying  from  0.16  to  0.33. 

The  tub-wheel  does  not  give  a  much  better  result.     A  verti- 
cal cylindrical  well  made  of  masonry  receives  the  water  from 
the  head  race  through  a  conducting  channel  A  (Fig.  12),  whose 
sides    incline    towards 
each  other  with  an  in- 
clination of  about  -   as 
5 

they  recede  from  the 
well ;  that  is  to  say, 
that  the  sides  make  an 
angle  of  11  or  12  de- 
grees with  each  other. 

One  side  is  also  tangent  to  the  circumference  of  the  well.  The 
wheel,  whose  axis  coincides  with  that  of  the  well,  consists  of  a 

certain  number  of  paddles  regularly  distributed  around  a  ver- 
5 


FIG.  12. 


68  TUB   WHEELS. 

tical  axle.  The  horizontal  section  of  the  paddles  presents  a 
slightly  curved  form,  having  its  concavity  towards  the  side 
from  which  the  action  of  the  water  comes ;  cut  by  a  cylinder 
concentric  with  the  well,  they  would  give  inclined  lines  more 
or  less  like  arcs  of  helices.  The  working  of  the  machine  is 
easily,  understood  :  the  water  comes  through  the  trough  A  with 
considerable  velocity,  endeavors  to  circulate  all  around  the 
well,  and,  meeting  the  paddles  in  its  road,  obliges  them  to  turn, 
as  well  as  the  axle  that  supports  them.  At  the  same  time,  the 
water  obeys  the  law  of  gravity,  passes  through  the  wheel  by 
means  of  the  free  space  between  the  paddles,  and  falls  into  the 
tail  race,  which  ought  to  be  a  little  lower.  "We  see  that  the 
water  must  undergo  a  good  deal  of  disturbance  in  entering  the 
wheel,  and,  moreover,  that  it  acts  upon  the  latter  for  too  short 
a  time  to  entirely  lose  its  relative  velocity.  Also  the  effective 
delivery,  sometimes  very  slight  and  about  0.15,  never  exceeds 
0.40. 

We  will  pass  over  these  primitive  machines  in  order  to  study 
others  more  perfect 


TURBINES. 


11.  Of  turbines. — The  principle  of  reaction  wheels,  such  as 
are  ordinarily  mentioned  in  Treatises  of  Physics,  has  long  been 
known  ;  but  it  appears  that  it  was  only  towards  the  beginning 
of  the  last  century  that  the  idea  was  entertained  of  making  use 
of  and  applying  it  to  the  construction  of  water  wheels  with  a 
certain  power.  Segner,  a  professor  at  Gottingen,  and  more 
recently  Euler,  in  1752,  made  them  the  object  of  their 
researches. 

In  1754  Euler  constructed  another  machine,  still  founded  on 
the  principle  of  reaction  wheels,  but  differing  from  them  in 
several  important  arrangements ;  this  machine  offers  the  most 
striking  resemblance  to  a  powerful  wheel  now  in  use,  called  in 
the  arts  Fontaine's  turbine,  from  the  name  of  the  skilful  con- 
structor, who  has  set  up  a  great  many  within  late  years,  besides 
extending  and  completing,  in  the  details  of  their  application, 
the  idea  first  advanced  by  Euler.  It  appears  that  this  kind  of 
wheel  was  not  much  used  until  towards  1824,  the  period  at 
which  the  question  was  again  studied  by  M.  Burdin,  engineer 
in  chief  of  mines,  who  constructed  a  similar  machine  which  he 
called  a  reaction  turbine.  In  the  years  following,  M.  Fourney- 
ron,  inspired  by  the  ideas  of  M.  Burdin,  established  some  tur- 
bines in  which  he  introduced  marked  improvements.  Since 
that  time  turbines  have  greatly  spread  and  multiplied ;  and 
there  exists  a  great  number  of  models  which  differ  more  or  less 
from  each  other. 


70  TURBINES. 

Without  undertaking  to  follow  up  more  completely  the  his- 
tory of  the  changes  successively  undergone  by  this  kind  of 
wheel,  we  shall  briefly  describe  the  three  principal  classes  into 
which  the  turbines  now  in  use  may  be  divided ;  then  we  will 
give  a  general  theory  for  them,  and  finally  we  will  mention 
some  details  to  which  a  particular  interest  is  attached. 

12.  Fourneyron* s  turbine. — The  essential  parts  of  this  tur- 
bine are  represented  in  (Fig.  13). 

The  water  from  the  head  race  A  descends  into  the  tail  race 
B  by  following  a  tube,  with  a  circular  horizontal  section,  of 
which  C  D  is  the  upper  opening.  This  tube,  which  is  perma- 
nently fixed,  rests  on  supports  of  timber  or  masonry;  it  is  pro- 
longed by  another  circular  cylinder  of  cast  iron  E  G I  F,  mov- 
able vertically,  which  can  be  lowered  more  or  less,  by  means  to 
be  presently  explained.  The  bottom  K  K'  K"  L"  I/  L  of  the 
tube  is  joined  to  a  hollow  cylinder  or  pipe  a  I  c  d,  supported  at 
its  upper  end ;  this  pipe  is  moreover  intended  to  keep  the  ver- 
tical shaft  0/*from  contact  with  the  water,  a  motion  of  rotation 
being  given  to  the  shaft  by  the  flow  of  the  water  due  to  its 
head.  In  fact,  we  see  that  if  the  water  reached  the  axle,  it 
would  be  necessary,  in  order  to  avoid  leakage,  to  make  this 
latter  pass  through  tightly  closed  packings,  which  would  occa- 
sion friction,  independently  of  that  caused  by  the  contact  of 
the  fluid.  The  arbor,  as  well  as  the  pipe,  are,  moreover,  con- 
centric with  the  tub. 

Between  the  bottom  G  I  of  the  cylindrical  sluice  E  G  I  F  and 
the  annular  plate  K  L,  there  is  an  opening  G  K,  I  L,  entirely 
around  the  perimeter  of  the  bottom,  through  which  the  water 
can  flow.  But  as  it  is  important,  as  we  shall  see,  that  the 
threads  should  not  flow  in  any  direction  whatever,  they  are 
guided  in  their  exit  by  a  certain  number  of  cylindrical  par- 
titions with  vertical  generatrices,  which  are  supported  by  the 


TURBINES. 


71 


Fio.  13. 


72  TURBINES. 

plate  K  L,  and  of  which  the  arrangement  is  sufficiently  well 
indicated  in  the  horizontal  section ;  amongst  these  directing 
partitions  or  guides,  some,  such  as  g  A,  are  joined  to  the  sides 
K'  K",  I/  L" ;  others,  such  as  i  &,  are  shorter,  in  order  to  avoid 
too  great  a  proximity  in  the  extremities  of  the  partitions 
towards  the  axle. 

"With  regard  to  the  opening  G  K,  IL,  the  turbine  proper 
is  included  between  the  two  annular  plates  or  crowns 
SRMN,  YTPQ;  these  plates  are  connected  together  by 
means  of  the  floats  of  the  turbine,  which  are  cylindrical 
surfaces  with  vertical  generatrices,  giving  in  horizontal  sec- 
tion a  series  of  curves,  such  as  I  m,  p  q,  &c. ;  the  lower  plate 
is  further  connected  with  the  arbor  by  a  surface  of  revolu- 
tion T  a  j3  P?  bolted  to  it,  so  as  to  form  a  perfectly  solid  whole. 
The  axle  rests  on  a  pivot  at  its  lower  end  ;  a  lever  7  £, 
moved  by  a  rod  §  s,  which  ends  at  a  point  easily  reached, 
allows  this  pivot  to  be  raised  a  very  little,  when  the  wear 
and  tear  of  the  rubbing  surfaces  has  produced  a  slight  settling 
of  the  axle. 

To  see  how  the  action  of  the  water  sets  the  machine  in 
motion,  let  us  suppose  first  that  the  arbor  is  made  fast :  then 
the  liquid  threads,  leaving  the  well  through  the  directing  par- 
titions, will  strike  against  the  concavity  of  the  floats ;  they  will 
thus  exert  a  greater  pressure  on  the  concave  than  on  the  con- 
vex portion  of  a  channel  such  as  Impq,  formed  of  two  con- 
secutive floats,  first  by  virtue  of  the  shock,  and  secondly  the 
curved  path  that  they  are  obliged  to  describe.  Hence  there 
would  result  a  series  of  forces  whose  moments,  relative  to  the 
axis,  would  all  tend  to  turn  the  system  of  the  floats  in  the 
direction  of  the  arrow-head  indicated  in  the  horizontal  section ; 
thus  there  will  be  produced  effectively  a  rotation  in  the  direc- 
tion indicated,  if  the  axle  be  allowed  to  turn,  even  in  opposing 


TURBINES.  73 

it  by  a  resistance  of  which  the  moment  should  be  inferior  to 
the  entire  moment  of  the  motive  forces. 

To  diminish  as  much  as  possible  the  loss  of  head  experienced 
by  the  liquid  molecules  during  their  passage  from  the  head 
race  to  the  wheel,  we  should  take  care  :  1st,  to  give  the  opening 
CD  a  sufficiently  great  diameter  and  to  round  off  its  edges; 
2d,  to  furnish  the  circular  sluice  with  wooden  appendages  G  G', 
I I',  placed  at  the  lower  portion,  and  having  their  edges  rounded 
off,  as  shown  in  the  figure.  We  shall  thus  sufficiently  avoid 
the  whirls  and  eddies  caused  by  the  successive  contractions  of 
the  threads  of  water.  The  wooden  packing,  moreover,  is  not 
continuous ;  it  is  composed  of  a  series  of  pieces,  each  occupying 
the  free  space  between  two  consecutive  partitions,  so  that  the 
sluice  can  be  lowered  without  any  hindrance  as  far  as  the  bot- 
tom K  K'  L  I/. 

"We  shall  see,  in  considering  the  general  theory  of  turbines^ 
how  the  other  conditions  essential  to  a  good  hydraulic  motor 
are  fulfilled. 

13.  Fontaine's  turbine. — Fig.  14  shows  a  general  section  of 
this  machine  by  a  vertical  plane.  A  piHar  or  vertical  metallic 
support  A  B  is  set  as  firmly  as  possible  into  the  masonry  form- 
ing the  bottom  of  the  tail  race ;  this  supports  at  its  upper  end 
A  a  hollow  cast-iron  axle  G  D  E  F,  which  surrounds  it ;  this 
axle  is  prolonged  above  by  a  solid  one,  upon  which  is  the  mech- 
anism for  transmitting  the  motion.  A  screw  and  nut  C  allows 
the  position  of  the  axle  to  be  regulated  in  a  vertical  direction. 
Nearly  at  the  level  of  the  water  in  the  tail  race  (or,  if  desirable, 
below  it)  is  placed  the  turbine  HIKLMNOP,  permanently 
fastened  to  the  bottom  of  the  hollow  axle;  it  is  comprised 
between  two  surfaces  of  revolution  concentric  with  the  vertical 
axis  of  the  system ;  these  surfaces  having  H  K  and  1 L  for 
meridian  lines  ;  in  the  intermediate  space  are  placed  the  floats, 


74: 


TURBINES. 


FIG.  14. 


TURBINES.  75 

which  receive  the  action  of  the  water,  and  at  the  same  time 
strengthen  the  two  surfaces.  The  water  comes  from  the  head 
race  a  to  the  floats  by  flowing  through  a  series  of  distributing 
canals,  of  which  the  quadrilaterals  QRHI,  STMN  represent 
the  sections.  These  channels  are  distributed  continuously  over 
an  annular  space,  directly  over  the  floats;  they  are  limited 
laterally  by  the  surfaces  QHTN, RISM;  the  space  between 
these  surfaces,  moreover,  remains  free,  except  the  volume  occu- 
pied by  the  directing  partitions,  which  divide  it  into  a  certain 
number  of  inclined  channels,  in  which  the  liquid  threads  move 
with  a  determinate  figure  and  direction. 

To  give  a  clear  idea  of  the  shape  of  the  directing  partitions 
and  floats,  let  us  suppose  a  section  made  by  a  cylinder  or  a 
cone,  concentric  with  the  axis  of  the  system,  passing  through 
the  middle  of  the  spaces  Q  R,  H  I,  K  L,  and  this  section  deve- 
loped on  a  plane.  The  developed  section  of  the  directing  sec- 
tions will  give  a  series  of  curves  such  as  c  d,  ef,  .  .  .  comprised 
in  a  straight  or  curved  row  ;  in  like  manner,  for  the  floats  of 
the  turbine,  we  shall  obtain  the  curves  dg,fh,  .  .  .  also  com- 
prised in  another  row.  These  curves  having  been  drawn  con- 
formably to  rules  which  we  shall  consider  further  on,  let  us 
suppose  reconstructed  the  cylinder  or  cone  that  had  been  devel- 
oped, and  let  us  conceive  the  warped  surfaces  generated  by  a 
right  line  moving  along  the  axis  and  on  each  of  the  curves  in 
question  successively ;  we  shall  in  this  way  have  determined 
the  surfaces  of  the  partitions  and  floats. 

It  is  deemed  unnecessary  to  describe  the  arrangement  for  the 
water  from  the  head  race  to  flow  into  the  tail  race  in  no  other 
way  than  through  the  channels  formed  by  the  directing  parti- 
tions ;  in  this  respect  the  figure  gives  sufficiently  clear  indica- 
tions. 

We  can,  as  in  the  case  of  Fourneyron's  turbine  (No.  12),  ac- 


76  TtfRBINES. 

count  for  the  direction  in  which  the  machine  should  turn  on 
account  of  the  action  of  the  water ;  and  which  is  that  of  the 
arrow-head  drawn  below  the  development  of  the  floats. 

14.  KaMin's  turbine.  —  Koecklin's  turbine,  of  which  the 
entire  arrangement  was  first  imagined  by  a  mechanic  named 
Jonval,  does  not  differ  essentially  from  Fontaine's  turbine, 
either  in  the  arrangement  of  the  floats  and  directing  partitions, 
or  the  mode  of  action  of  the  water.  The  most  noticeable  dif- 


Q^X^Si^SSS^ssi^^ 

FIG.  15. 

ference  consists  in  this,  that  the  turbine  is  above  the  level  of 
the  water  in  the  tail  race,  as  shown  in  (Fig.  15),  a  vertical  sec- 
tion of  the  apparatus.  The  directing  partitions,  set  in  an 


TURBINES.  77 

animlar  space,  of  which  the  trapczoids  Q  R  H  I,  S  T  M  K, 
indicate  sections,  are  fastened  to  a  kind  of  cast-iron  socket, 
which  enjbraces  the  axle  A  B,  without,  however,  forming  part 
of  it,  or  pressing  it  hard  ;  they  form  a  set  of  inclined  channels, 
through  which  the  water  from  the  head  race  flows  and  reaches 
the  floats  of  the  turbine,  placed  immediately  below,  as  in  Fon- 
taine's turbine.  These  floats  are  fastened  to  another  socket, 
which  is  bolted  to  the  axle ;  they  occupy  the  annular  space 
H  I  K  L,  M  E"  O  P. 

The  inclined  channels  included  between  two  consecutive 
floats  or  partitions  are  limited  on  the  outside  by  a  fixed  tub  of 
cast-iron,  resting  on  the  edges  of  a  well  of  masonry  ;  it  forms  a 
surface  of  revolution  about  the  axis  A  B,  having  Q  H  K  D  for 
a  meridian  section.  At  the  bottom  of  this  tub  are  found  a  cer- 
tain number  of  arms  which  support  a  centre  piece,  on  which  is 
placed  the  pivot  of  the  revolving  shaft. 

The  water  having  left  the  turbine,  by  the  apertures  K  L, 
O  P,  flows  into  the  tail  race  by  descending  througli  the  mason- 
ry well,  and  then  passing  into  an  opening  which  we  may  con- 
tract, or,  if  need  be,  close  at  will,  by  means  of  a  sluice  Y. 

The  situation  of  the  turbine  above  the  level  of  the  water  in 
the  tail  race  allows  it  to  be  easily  emptied,  and  herein  lies  its 
principal  advantage ;  for  this  purpose  we  have  merely  to  leave 
the  sluice  of  the  tail  race  open,  and  to  prevent  the  water  reach- 
ing the  distributing  channels  Q  K  H  I,  S  T  M  N.  We  can  then 
visit  the  machine  and  make  the  necessary  repairs.  Besides, 
the  height  included  between  the  horizontal  plane  K  L  O  P  and 
the  level  of  the  water  in  the  tail  race  should  not  be  considered 
as  a  loss  of  head,  because  it  belongs  to  a  diminution  of  pressure 
on  the  water  which  leaves  the  turbine,  and  we  shall  see  by  the 
general  theory,  now  to  be  investigated,  that  this  causes  an 
exact  compensation. 


78  TURBINES. 

15.  Theory  of  the  three  preceding  turbines. — A  complete 
theory  of  the  turbines  that  we  have  just  summarily  described 
ought  to  include  first  the  solution  of  the  following  general 
problem.  Having  given  all  the  dimensions  of  a  turbine,  its 
position  as  regards  the  head  and  tail  races,  and  finally  its  angu- 
lar velocity,  to  determine  the  volume  of  water  that  it  expends, 
and  the  dynamic  effect  of  the  head,  we  should  then  seek  the 
conditions  necessary  to  make  the  effective  delivery  for  a  given 
head  and  expenditure  of  water  a  maximum. 

But  we  shall  not  treat  the  question  in  such  general  terms. 
In  order  to  simplify  the  researches  with  which  we  are  to  be 
employed,  we  shall  allow  in  what  follows  that  all  the  dimen- 
sions have  been  chosen  and  the  arrangements  made,  so  that  the 
turbine  may  fulfil  in  the  best  possible  way  the  conditions  for  a 
good  hydraulic  motor.  Then,  from  the  pond  to  the  exit  from 
the  directing  partitions,  care  will  have  been  taken  to  avoid 
contractions  and  sudden  changes  in  direction  of  the  threads ; 
to  have  smooth  and  rounded  surfaces  in  contact  with  the  water 
that  flows  through,  in  order  that,  in  this  first  portion  of  its 
passage,  it  may  meet  with  no  sensible  loss  of  head.  At  the 
point  of  entrance  into  the  turbine,  the  water  possesses  a  certain 
relative  velocity ;  matters  will  be  so  arranged  that  this  velocity 
shall  be  directed  tangentially  to  the  first  elements  of  the  floats, 
in  order  that  no  shock  or  violent  disturbance  may  follow.  This 
is  a  condition  that  it  is  possible  to  fulfil  by  choosing  a  suitable 
velocity  for  the  wheel,  as  well  as  proper  directions  for  the  floats 
and  partitions  where  they  join.  Finally,  as  the  water  leaves 
the  turbine  in  every  direction  about  a  circumference,  and  as  it 
is  hardly  possible  to  prevent  the  absolute  velocity  which  it 
then  possesses  being  used  up  as  a  dead  loss  in  producing  eddies 
in  the  tail  race,  we  shall  suppose  that  we  have  taken  care  to 
make  this  velocity  small.  All  this  combination  of  circum- 


TTJRBINES.  79 

stances  will  considerably  simplify  our  calculations,  by  allowing 
us  to  neglect  in  them,  without  a  very  sensible  error,  the  differ- 
ent losses  of  head  experienced  by  the  water  up  to  the  point  of 
its  exit  from  the  turbine,  a  loss  of  which  the  analytical  expres- 
sion, more  or  less  complicated,  would  overload  our  formulas 
and  make  them  much  less  manageable.  Only,  it  should  be 
understood  that  our  results  will  be  exclusively  applicable  to  the 
case  in  which  the  machine  works  according  to  the  conditions 
for  a  maximum  effective  delivery. 

Thus  granted,  let  us  call 

v  the  absolute  velocity  of  the  water  when  it  leaves  the 
directing  partitions  and  enters  the  turbine ; 

u  its  impulsive  velocity  and  w  its  relative  velocity,  at  the 
same  point,  with  respect  to  the  turbine  taken  for  a  system  of 
comparison ; 

v',  uf  and  w'  the  three  analogous  velocities  for  the  point  at 
which  the  water  leaves  the  wheel ; 

p  and  p'  corresponding  pressures  at  these  two  points ; 

pa  the  atmospheric  pressure ; 

r  and  r'  the  distances  of  the  same  two  points  from  the  axis 
of  rotation  of  the  system*  ; 

H  the  height  of  the  head,  measured  between  the  level  of  the 

(*)  These  two  distances  are  often  equal  in  the  Fontaine  and  Koecklin  tur- 
bines ;  but  they  should  differ  in  a  marked  degree  in  Fourneyron's  turbine. 
Moreover,  it  should  be  observed  that,  in  the  Fontaine  and  Krecklin  turbines, 
the  outlets  of  the  water,  from  the  guide  curves,  have  a  certain  dimension  per- 
pendicular to  the  axis  of  rotation  ;  the  lengths  r  and  r',  be  it  well  understood, 
should  be  referred  to  mean  points  of  these  outlets.  Thus  in  Fig.  15,  for 
example,  r  should  be  the  mean  of  U  M  and  UN;  r',  in  the  same  way,  should 

be  equal  to ~ .     M  N  and  0  P  should  also  be  so  taken  as  to  be  small 

relatively  to  r  and  /,  in  order  that  the  consideration  of  a  single  thread  of 
water  only  may  not  cause  an  appreciable  error. 


80  TURBINES. 

water  in  the  pond  and  tail  race,  supposed  to  be  sensibly  at 
rest; 

h  the  depth,  positive  or  negative,  of  the  point  of  entrance  of 
the  water  into  the  interior  of  the  turbine,  below  the  level  of 
the  tail  race ; 

hr  the  height  the  water  descends  during  its  motion  in  the 
interior  of  the  turbine,  a  quantity  equal  to  zero  when  M.  Four- 
neyron's  arrangements  are  adopted  ; 

n  weight  of  the  cubic  metre  of  water. 

Now,  all  the  intervals  between  two  consecutive  directing 
partitions  being  considered  as  a  first  system  of  curved  channels, 
and  all  the  intervals  between  two  consecutive  floats  as  a  second 
system.  For  the  first  system  of  these  channels,  represent  by, 

£  the  acute  angle  under  which  they  cut  the  plane  of  the 
orifices  that  terminate  them  at  the  distance  r  from  the  axis ; 
this  angle  /3  is  also  that  at  which  the  circumference  2  if  r  is  cut 
by  the  partitions ; 

b  the  depth  or  breadth  of  the  orifices  of  which  we  have  just  been 
speaking,  measured  perpendicularly  to  the  circumference  2  *  r. 

For  the  second  system  : 

6  the  angle  made  by  the  plane  of  the  orifices  of  entrance 
with  the  direction  of  the  floats,  or,  what  amounts  to  the  same 
thing,  the  angle  of  these  floats  with  the  circumference  2  *r  /•, 
to  which  we  will  give,  moreover,  a  direction  opposed  to  the 
velocity  u,  the  direction  of  the  float  being  taken  in  that  of  the 
relative  motion  of  the  water ; 

7  the  acute  angle  at  which  the  channels  cut  their  orifices  of 
exit,  or,  in  other  words,  the  angle  of  the  floats  with  the  cir- 
cumference 2  *  rf  about  which  the  above-mentioned  orifices  are 
distributed ; 

V  the  depth  or  breadth  of  the  orifices  of  exit,  measured  per- 
pendicularly to  the  circumference  2  *  rf. 


TURBINES.  81 

The  question  now  is  to  establish  the  relations  between  all 
these  quantities,  under  the  supposition  that  the  conditions  of 
the  maximum  effective  delivery  are  satisfied.  For  that  pur- 
pose we  shall  first  follow  the  motion  of  a  molecule  of  water 
along  its  path  between  the  two  races,  and  write  out  the  equa- 
tions furnished  by  Bernouilli's  theorem. 

Between  a  point  of  departure  taken  in  the  pond,  where  there 
is  no  sensible  velocity,  and  the  point  of  exit  at  the  extremity 
of  the  directing  partitions,  there  is  a  head  expressed  by  H  -f-  h 

nn      _   £) 

H  —  *2  —  £-•  the  velocity  being  v,  we  have  then,  under  the 


supposition  of  a  loss  of  head  that  can  be  disregarded  between 
the  two  points  in  question, 

«*  =  2  g  (H  +  A  +  •=£*=£-  ----  (1) 


The  water  then  moves  along  the  floats  of  the  turbine  with  a 
velocity  at  first  equal  to  w  and  afterwards  to  wf  :  in  this  second 
period,  if  we  call  w  the.  angular  velocity  of  the  machine,  we 
know,  applying  Bernouilli's  theorem  to  the  relative  motion, 

there  must  be  added  to  the  real  head  h'  -f  —  —  £-,  a  fictitious 

w«  /'  _  w-  ^          „  »  _  „• 

head  -  -  ---  ,  or  —  -  --  ;   still,  neglecting  the  losses 
^  9  *  9 

of  head,  which  is  approximately  admissible  when  the  water 
enters  with  a  relative  velocity  tangent  to  the  floats,  we  shall 
then  have  to  place 

w"  -  w*  =  2  g  (V  +^^)  +  ™/3  -  u?  •  •  •  •  (2) 

When  the  turbine  is  immersed  and  h  is  positive,  the  point  of 
exit  of  the  water  is  found  at  a  depth  A  -f  h'  below  the  level  of 
the  tail  race  ;  besides,  as,  for  the  maximum  effective  delivery, 
the  water  must  go  out  with  a  slight  absolute  velocity,  we  can 


82  TURBINES. 

without  material  error  admit  that  the  pressure  varies,  in  the 
tail  race,  according  to  the  law  of  hydrostatics,  which  gives 

&  +  h  +  h'  =  g (3) 

This  relation  is  also  true  in  Kcecklin's  turbine,  although 
A  H-  A'  becomes  negative,  provided  that  the  well  placed  below 
the  turbine  and  the  orifice  by  which  it  communicates  with  the 
tail  race  are  sufficiently  large ;  for  then  the  water  will  take  up 
in  it  but  a  slight  velocity  and  can  there  be  considered  as  in 
equilibrio ;  the  pressure  p'  would  then  be  less  than  pa  by  a 
quantity  represented  by  the  depth  —  (h  +  A'),  as  equation  (3) 
shows.  We  could  also  preserve  it,  if  the  lower  plane  of  the 
turbine,  constructed  according  to  one  of  the  first  two  systems, 
were  on  the  same  level  as  the  water  in  the  tail  race  :  we  would 
then,  in  fact,  have pa  =p'  and  A  +  h'  =  <?,  the  quantities  A  and 
A'  being  with  contrary,  signs,  or  else  both  equal  to  zero,  accord- 
ing to  whether  we  are  considering  Fontaine's  or  Fourneyron's 
turbine. 

The  incompressibility  of  water  will  furnish  us  with  the 
fourth  equation,  showing  that  the  volume  of  water  that  has 
flowed  between  the  directing  partitions  is  equal  to  that  which 
leaves  the  turbine.  The  distributing  orifices,  left  free  between 
the  partitions,  occupy  a  total  development  2  «  r  (excepting  the 
slight  thickness  of  the  partitions)  and  a  breadth  £,  from  whence 
there  results  a  surface  2  *  J  r  ;  as  they  are  cut  by  liquid  threads 
moving  with  the  velocity  v,  at  an  angle  /?,  we  have  for  the  first 
expression  of  the  volume  Q  expended  in  a  unit  of  time 

Q  =  2  *  5  r  sin  ft.  v. 

In  like  manner,  the  orifices  of  exit  at  the  extremity  of  the 
floats  have  a  total  development  2  if  r',  a  breadth  Z>',  a  surface 
2  if  1)'  r' ,  and  they  are  cut  by  the  threads  moving  with  a  rela- 
tive velocity  wf  at  an  angle  7' ;  then 

Q  =  2  ic  V  r'  sin  7.  w'. 


TURBINES.  83 

Strictly  speaking,  on  account  of  the  thickness  of  the  floats 
or  partitions,  these  two  expressions  for  the  value  of  Q  ought  to 

undergo  a  slight  relative  reduction  of—  or  -— ;  but  in  all  cases, 

Jo       oO 

the  reduction  being  the  same  for  both,  we  shall  have  by  the 
equality  of  the  values  of 

2l     IT 

b  r  v  sin  /3  =  ~b'  rf  w'  sin  7 (4) 

The  three  following  relations  are  in  a  certain  degree  geo- 
metrical. 7 

Let  us  represent  (Fig.  16)  a  float  B  C  and  a  directing  par- 
titicfn  A  B ;  a  liquid  mole- 
cule having  followed  the 
path  A  B  arrives  at  B  with 
an  absolute  velocity  v,  and  a 
velocity  w  relatively  to  the 
turbine  which  itself,  at  the 
point  B,  possesses  the  velo- 
city u.  This  last  being  what 

is  called  the  propelling  velocity,  we  know  that  v  is  the  diagonal 
of  the  parallelogram  constructed  on  u  and  w  ;  and  as  the  angle 
between  v  and  u  is  exactly  /?,  the  triangle  BUY  will  give 

UT2  =  FtT+TTf  -  2BU.  FY  cos  /8, 

that  is, 

wz  =  it?  4-  v*  —  2  u  v  cos  /3 (5) 

In  like  manner  the  liquid  molecule,  after  having  traversed, 
relatively  to  the  turbine,  the  path  B  C,  arrives  at  C  with  the 
velocity  w' ',  which,  taken  as  a  component  with  the  propelling 
velocity  u' ',  gives  the  absolute  velocity  v';  then  the  angle  7 
being  the  supplement  of  that  made  by  u'  and  w ',  we  shall  have 

v'*  =  u"  +  w'*  —  2  u'  w'  cos  7 (0) 

6 


84:  TUKBINES. 

On  the  other  hand,  the  velocities  u  and  u'  belong  to  two  points 
of  the  turbine  situated  respectively  at  the  distances  r  and  r'  of 

the  axis  of  rotation,  we  have  then  —  =  —  ,  or 

u'  r  =  u  rf  .....  (7) 

There  still  remains  to  express  two  conditions  necessary  for 
obtaining  the  best  effective  delivery.  It  is  necessary  first,  at 
the  point  B,  that  w  be  directed  tangentially  to  the  floats  B  C 
without  which  there  would  be  a  sudden  change  of  relative 
velocity,  whence  would  result  disturbance  and  a  loss  of  head 
that  we  have  not  considered.  Now  the  angle  between  w  and  u 
is  the  supplement  of  0,  hence  the  triangle  BUY  gives 

B_U    _  sin  B  Y  U    _  sin(BUY  +  Y  B  U) 
B  Yzr  sinBUY1  sin  BUY 

or 

u        sin    6  +  3  ,, 


v  sn 

It  is  then  necessary  that  the  absolute  velocity  vr  possessed  by 
the  water  on  leaving  the  turbine  should  be  very  slight,  since 

v'* 

-  —  enters  in  the  loss  of  head  (No.  3)  :  this  condition  is  suf- 

ficiently satisfied  by  taking  the  angle  7  small,  and  placing 

u'  =  w';  ....  (9) 

for  then  the  parallelogram  C  U7  Y'  W  is  changed  into  a  lozenge, 
very  obtuse  at  one  angle  and  very  acute  at  the  other,  and  the 
diagonal  joining  the  obtuse  vertices  is  short  ;  in  other  words, 
the  velocities  u'  and  w'  are  equal,  and  almost  directly  opposed, 
which  makes  their  resultant  very  small. 

We  have  thus  obtained,  in  all,  nine  equations  between  six- 
teen Tariable  quantities  in  a  turbine,  namely  — 

-six  velocities  w,  t>,  w,  u'  ,  v',  w\ 

two  pressures  p,  p', 


TURBINES.  85 

r     I 

two  ratios  —,  ^7, 

three  altitudes  H,  A,  A', 

three  angles  /3,  y,  6. 

These  equations  will  serve  us  for  solving  two  distinct  problems, 
which  may  be  thus  stated  :  1st,  having  given  a  turbine  and  all 

r    I 

its  dimensions  (that  is  to  say,  the  eight  quantities  /3,  y,  d,  —  -  —  — 

H,  A,  A'),  to  show  the  conditions  these  dimensions  must  satisfy, 
in  order  that  the  turbine  may  work  with  the  maximum  effective 
delivery  —  that  is  to  say,  so  that  the  nine  above  equations  may 
obtain  ;  and,  under  the  supposition  that  these  conditions  are 
fulfilled,  to  show  the  most  suitable  velocity  of  the  turbine,  as 
well  as  the  expenditure  of  water  corresponding  to  this  velocity, 
its  effective  delivery,  and  its  dynamic  effect  ;  2d,  having  given 
the  expenditure  and  the  height  of  a  head,  to  establish  under 
this  head  a  turbine  with  the  best  conditions. 

The  first  question  involves  eight  unknown  quantities,  which 
are  u,  v,  w,  u'  ,  vf,  wr,  p,  p'  ;  the  elimination  of  these  unknown 
quantities  between  the  nine  equations  will  then  give  an  equa- 
tion of  condition  to  be  satisfied  by  the  dimensions  of  <the  ma- 
chine —  an  equation  to  which  we  shall  have  to  add  two  others 
in  order  to  show  that  the  pressures^?  and  ^  are  essentially  posi- 
tive. The  following  calculation  has  for  its  object  to  bring  out 
these  three  conditions,  and,  at  the  same  time,  to  give  the  value 
of  the  unknown  quantities. 

Adding  equations  (1),  (2),  and  (5),  member  to  member,  there 
obtains 


n  =  2  g  ( 


H  +  h  +  hf  +LZL        +  u*  -  2  u  v  cos  ft 


or,  considering  (3)  and  (9), 

u  v  cos  £  =  g  H  .  .  .  .  (10) 


86  TUBBIKES. 

The  combination  of  equations  (4)  and  (9)  readily  give 

1}  v  r  sin  j8  =  ~b'  u'  r'  sin  7  ;  .  .  .  .  (11) 

multiplying  equations  (7),  (10),  and  (11),  member  by  member, 
there  obtains 

tf  I  r*  sin  jS  cos  /3  =  g  H.  V  rf*  sin  7  ; 

whence  one  of  the  unknown  quantities 

<f  =  gnvjr^™l—  ....(12) 

5  7*a  sin  /3  cos  /3 

02  H2 
We  have  besides,  from  equation  (10),  u*  =  -j—    •-  ;  whence 

V   COS  ]8 


and  from  equation  (7) 

„«          H  &  tan* 
Jr  sin  7 
To  obtain  -y7,  we  will  first  make  w'  =  u'  in  equation  (6),  which 

will  give 

v"  —  2  u'z  (1  —  cos  7), 

and  consequently  from  the  value  of  u'  (14), 


Knowing  w7  we  obtain  w\  and,  if  w  were  required,  we  could 
easily  obtain  it  by  substituting  in  equation  (5)  the  values  of  v 
and  u.  Thus,  all  the  velocities  may  be  considered  as  known  ; 
we  could  deduce  from  them  the  angular  velocity  w  with  which 
the  turbine  must  move,  when  it  is  working  with  the  conditions 
of  the  maximum  effective  delivery  ;  we  would  then  have  prac- 
tically 

_  u  _  u' 
r       r'' 

The  corresponding  expense  Q  has  for  its  value  2  *  ~b'  wf  r'  sin  7, 
or,  substituting  in  place  of  w'  the  value  of  its  equal  u'  from 
equation  (14), 


TURBINES.  87 


Q  =  2*V  VlV  V  g  H  tan  /3  sin  /,  ....  (16) 
a  formula  whose  second  member  we  should  probably  have  to 
multiply  by  a  number  less  than  unity,  in  order  to  take  into 
consideration  the  space  occupied  by  the  floats,  and  also  to  com- 
pensate for  the  influence  of  the  losses  of  head  neglected  in  the 
calculation. 

Let  us  now  seek  the  three  equations  of  condition  to  be  satis- 
fied by  the  dimensions  of  the  turbine.  First,  by  dividing  equa- 
tions (13)  and  (12),  member  by  member,  and  extracting  the 
square  root  of  the  quotient,  we  will  find 

u  _  5  ra  sin  /3 
v      V  rn  sin  7' 

and  by  reason  of  eq.  (8) 

sin   &  +  j3  _  ~b  r*  sin  /3  1  ,._ 

~a~ 


sn 

this  is  the  condition  obtained  by  eliminating  eight  unknown 
quantities  between  nine  equations.  There  remains  yet  to  ex- 
press that^>  >  0,  j/  >  0.  As  to  this  last  condition,  we  see  from 
(3)  that  it  is  itself  satisfied  for  Fourneyron's  and  Fontaine's 
turbines,  by  supposing  that  they  are  on  a  level  with  the  water 
in  the  tail  race  or  below  it,  as  we  have  admitted  in  the  preced- 
ing calculations  ;  because  then  h  +  ti  is  positive,  and  we  have 
p'  >  pa.  In  Koecklin's  turbine,  on  the  contrary,  the  bottom 
of  the  turbine  is  in  reality  above  the  level  of  the  water  in  the 

Pr 
tail  race,  by  a  positive  height  expressed  by  —  (h  +  hf)  ;  ~  has 

for  its  value  —  ,  or  10m.33  less  this  height  ;  then  it  is  absolutely 

necessary  to  have 

-  (h  +  h')  <  10m.33, 

and  perhaps  even,  on  account  of  neglected  losses  of  head,  it 
would  be  well  to  place 


88  TURBINES. 

-  (A  +  AO  <  «-.  ,  ----  (18) 

in  order  to  make  perfectly  sure  of  the  continuity  of  the  liquid 
column  in  the  cylindrical  well,  above  which  the  turbine  is 
found.  As  to  the  pressure  j?,  it  will  be  found  from  eq.  (1)  after 
substituting  in  it  for  v*  its  value  in  eq.  (19),  that  is 


(19) 


n  n  r    sin  p  cos  /a 

The  second  member  of  this  equation  should,  of  course,  be 
greater  than  zero  ;  but  we  may  assign  it  a  higher  limit.  In 
fact,  if  we  examine  the  arrangement  of  the  different  systems  of 
turbines,  we  see  that  there  is  always  an  indirect  communication 
between  the  distributing  orifices,  situated  at  the  end  of  the 
directing  partitions,  either  with  the  tail  race,  or  with  the  exter- 
nal air.  This  communication  is  effected  by  the  play  necessarily 
left  between  the  turbine  proper  and  the  distributing  orifices. 
When  it  takes  place  with  the  tail  race,  p  cannot  differ  much 
from  the  hydrostatic  pressure^  +  n  A,  which  would  take  place 
in  a  piezometric  column  communicating  with  this  race,  and  at 
the  height  of  the  point  of  entrance  of  the  water  above  the 
turbine  ;  otherwise  there  would  be,  on  account  of  the  play  that 
we  have  just  spoken  of,  either  a  sudden  gushing  out,  or  suction 
of  the  water,  which  would  produce  a  disturbance  in  the  motion. 
When  it  is  with  the  atmosphere,^?  must,  for  a  similar  reason, 
be  equal  to^?a.  It  is  then  prudent,  in  the  first  case,  to  impose 
the  condition  that  the  two  terms  in  which  the  factor  H  appears, 
eq.  (19),  should  nearly  cancel  each  other,  or,  designating  by  ~k 
a  number  that  differs  little  from  unity,  to  make 
Vrf*  siny 

K    -      -r  -  -      -  --  ,     .      .      .     .     (  40) 

o  r   sin  /?  cos  /? 
k  is  moreover  rigorously  subjected  to  the  condition  that 

h  +  &  +  H  (1  -  ft) 


TURBINES.  89 

should  be  positive.     And  in  like  manner,  for  the  second  case, 
we  should  establish  the  condition 

H  0-  -  1TT  o-^-^)  +  h  =  h"  '  '  '  '  (20  bis) 

2 


1TT  o-- 

b  r2    2  sm  &  cos 

h"  being  a  very  small  height. 

Again  it  might  be  proposed,  for  a  turbine  known  to  be  work- 
ing with  the  maximum  effective  delivery,  to  seek  this  delivery 
as  well  as  the  dynamic  effect.  As  we  are  supposing  that  all 
losses  of  head,  other  than  that  due  to  the  velocity  of  exit  -y', 
may  be  disregarded,  the  head  that  is  turned  to  account  will  be 

TT      «" 

T? 

and  consequently  the  productive  force  /*  will  be  expressed  by 
H-    -- 

a  -    "        -ll  -  1  -     -*L- 

H  2^H' 

or  replacing  v'*  by  its  value 
J    tan  /8 


The  dynamic  effect  Te  would  be  obtained  by  finding  the  pro- 
duct f*  n  Q  H  of  the  effective  delivery  by  the  absolute  power 
of  the  head  ;  then  we  would  have,  from  eqs.  (16)  and  (21) 


Te  =  n  II  V'g  H.     2  *  r'  V I  V.      V  tan  /s  sin 

(,         I     tan  /3  )  ^ .  •  •  •  (22) 

x   \  1 r     .         (1  —  cos  7)  f 

(  Z>     sm  7 

Thus  have  we  now  solved  the  first  of  the  two  general  prob- 
lems proposed.  When  we  take  up  the  second,  which  consists 
in  setting  up  a  turbine  for  a  given  head,  Q  and  H  become  the 
known  quantities,  and  we  have,  between  the  nine  quantities 
/s,  7,  6,  r,  r',  &,  &',  A,  A7,  which  define  the  unknown  dimensions, 
only  equations  (16),  (IT),  (20),  or  (20  Us\  to  which  must  be 


90  TURBINES. 

added  (if  a  Koecklin's  turbine  is  in  question)  the  inequality 
(18) ;  still  this  inequality  leaves  a  certain  margin ;  and  it  is  the 
same  with  equations  (20)  and  (20  bis),  because  the  quantities  k 
and  Ji"  have  not  a  definite  value.  It  appears,  then,  that  the 
problem  is  very  indeterminate,  and  that  we  may  assume  almost 
all  the  above-mentioned  dimensions  arbitrarily ;  however,  the 
following  remarks  impose  restrictions  that  it  will  be  well  to 
keep  in  mind. 

16.  Remarks  on  the  angles  /3,  7,  d,  and  on  the  dimensions  J, 
V ,  r,  r',  A,  A'. — If  we  only  considered  the  expression  for  the 
effective  delivery  eq.  (21),  we  should  be  tempted  to  make  one 
of  the  angles  /3  or  7  equal  to  zero;  the  theoretical  effective 
delivery  would  then  become  practically  equal  to  unity.  But 
we  see  that  the  expenditure  Q  would  reduce  to  zero,  as  well  as 
the  dynamic  effect  Te :  hence  the  value  zero  is  not  admissible 
for  either  of  these  angles. 

Making  7  very  small,  the  channels  formed  by  two  consecu- 
tive floats  would  be  very  much  narrowed  at  the  point  of  exit 
of  the  water;  the  water  would  flow  with  difficulty  through 
these  narrow  passages,  and  there  would  be  danger  of  its  not 
following  exactly  the  sides  of  the  floats,  which  would  occasion 
eddies  and  losses  of  head.  On  the  other  hand,  a  large  value 
for  7  would  diminish,  very  likely,  the  effective  delivery  too 
much.  Between  these  two  points  to  be  avoided,  experience 
gives  a  value  of  20  or  30  degrees  as  affording  satisfactory 
results. 

As  to  the  angle  /3,  besides  the  reason  already  given,  there  is 
still  another  for  not  making  it  zero :  this  second  reason  is  that 
from  equation  (19) p  would  be  negative  for  ft  =  o  and  /3  =  90°. 
We  must  not  then  approach  too  closely  to  zero  or  to  90° ;  the 
limits  from  30  to  50  degrees  have  been  advised  by  some 
experts ;  but  there  is  none  that  is  absolute. 


TURBINES.  91 

Let  us  suppose  that  we  are  about  to  apply  equation  (20) ; 
multiplying  it,  member  by  member,  by  equation  (17)  we  find 
k  sin  (6  +  j8)  1 

sin  6  2  cos  /3 ' 

whence 

1       ^  _  2  cos  /3  sin  (&  +  /3)  —  sin  0 
&  sin  6 

or,  by  developing  the  sin  (6  +  /3), 

2  cos  /3  sin  (d  +  /3)  —  sin  6  =  sin  6  (2  cos3  /3  —  1)  +  2  sin  /3  cos  /8  cos  4 
=  sin  6  cos  2  /3  -f  cos  6  sin  2  /3 

=  sin  (2  /S  +  d)  ; 
we  can  then  write 

dn^_+0        1       L  .  (23) 

sin  0  & 

We  have  previously  seen  that  k  should  be  a  number  very  near 
unity ;  it  follows  that  sin  (2  j3  +  6)  should  be  small,  and  conse- 
quently that  2  j8  +  6  should  differ  but  little  from  180  degrees. 
If,  for  example,  we  took  /3  about  45  degrees,  6  would  be  about 
a  right  angle.  Besides,  it  is  not  well  to  have  6  greater  than 
90° ;  for,  if  we  refer  to  Fig.  16,  we  see  that  if  6  be  obtuse  the 
floats  should  have  a  form  like  B'  C',  presenting  a  considerable 
curvature;  and  experience  shows  that  in  a  very  much  curved 
channel  the  water  meets  with  a  greater  loss  of  head,  all  other 
things  being  equal :  the  liquid  molecules  then  tend  to  separate 
from  the  convex  portion,  which  gives  rise  to  an  eddy.  We  see 
also  in  Fig.  16  that,  in  taking  4  very  acute,  the  side  Y  U  of  the 
triangle  B  U  Y — that  is,  the  relative  velocity  at  the  entrance, 
would  tend  to  become  more  or  less  great,  which  would  be  hurt- 
ful, since  the  friction  of  the  water  on  the  floats  would  be  in- 
creased. Hence  6  should  be  an  acute  angle,  but  at  the  same 
time  almost  a  right  angle :  we  might  make  it  vary,  for  example, 
between  80  and  90  degrees. 

If  it  be  equation  (20  Ids)  and  not  equation  (20)  that  we  have 


92  TURBINES. 

to  apply,  the  same  reasons  obtain  for  taking  &  acute  and  nearly 
90  degrees  ;  but  the  sum  2/3  +  4  need  no  longer  differ  much 
from  180  degrees. 

After  having  fixed  the  values  of  0,  /,  and  d,  we  will  find  from 

equation  (17)  the  ratio        /a,  whence  we  can  obtain  either  —  or 

M 

—7,  the  other  having  been  assumed. 

It  is  in  favor  of  the  effective  delivery  to  have  -p  less  than 

unity,  as  formula  (21)  shows  ;  it  must  not,  however,  be  greater 
than  the  difference  b'  —  5,  and  must  be  proportional  to  the 
length  of  the  floats,  in  order  that  the  channels  between  two 
consecutive  floats  may  not  be  emptied  too  rapidly,  because  this 
emptying  would  give  rise  to  a  loss  of  head.  We  can  impose 

the  condition  that  V  —  ~b  should  be  less  than  —  the  length  of 

the  floats. 

rf 

As  we  have  already  said,  the  ratio  —  is  often  taken  equal  to 

r 

unity  in  Fontaine's  and  Kcecklin's  turbines,  but  it  is  neces- 
sarily greater  than  unity  in  Fourneyron's  turbine.  If  it  be 
taken  different  from  unity,  we  must  not,  except  for  particular 
reasons,  increase  the  difference  r'  —  71,  or  r  —  r',  for  we  should 
thus  lengthen  the  floats  and  in  crease  friction.  In  Fourneyron's 

turbine  —  varies  ordinarily  between  1.25  and  1.50. 

The  height  A',  from  which  the  water  descends  into  the  inte- 
rior of  the  turbine,  is  always  zero  in  Fourneyron's  turbines  ;  in 
the  other  two  systems  it  is  so  taken  that  the  floats  may  be  suf- 
ficiently, but  not  too  long,  regard  being  had  to  the  difference 
V  —  5.  As  to  the  height  A,  if  it  be  a  question  of  a  turbine  of 


TURBINES.  93 

Kcecklin's,  it  is  fixed  according  to  local  circumstances,  the  in- 
equality expressed  in  (18)  being  considered ;  if  it  be  of  one  of 
Fourneyron's  or  Fontaine's,  we  so  arrange  matters  that  its 
lower  plane  may  be  on  the  same  level  as  the  waters  in  the  tail 
race  when  at  their  minimum  depth. 

Finally,  M.  Fourneyron  recommends  giving  the  circular  sec- 
tion of  the  tub,  in  which  the  directing  partitions  of  his  turbines 
are  placed,  a  surface  at  least  equal  to  four  times  the  right  sec- 
tion of  the  distributing  orifices,  in  order  that  the  fluid  threads 
may  easily  pass  from  the  vertical  to  the  horizontal  direction, 
which  they  must  have  at  their  point  of  exit.  With  the  nota- 
tion employed  in  (No.  15),  we  can  write 

*  ?>a  >  4.  2  it  r  b  sin  /3, 
or  else 

r  >  8  b  sin  /3,  .  .  .  .  (24) 

the  sign  >  not  being  exclusive  of  equality. 

Let  us  now  show,  by  two  examples,  how  we  shall  be  enabled, 
by  means  of  these  considerations,  to  determine  the  dimensions 
of  a  turbine  to  be  established. 

17.  Examples  of  the  calculations  to  be  made  for  constructing 
a  turbine. — Let  it  first  be  determined  to  establish  a  Fourney- 
ron turbine  with  the  following  data  : 

Height  of  fall,  H  =  6m.OO. 

Yolume  expended  per  second,  Q  =  lmc.50. 

The  absolute  power  of  the  head  is  1500  x  6kgm  =  9000  kilo- 
grammetres  per  second,  or  120  horse-power. 

Since  the  angle  7  is  not  fixed  theoretically,  we  will  take  it 
(No.  16)  equal  to  25  degrees ;   we  will  also  make  k  =  1  (*), 
which  secures  that  p  shall  be  positive  (No.  15) ;  finally,  we  will 
take  6  =  90°.     Equation  (23)  then  gives 
sin  (2  ft  +  6)  =  0, 
whence 


94  TURBINES. 

2  /3  +  d  =  180°  and  0  =  45°. 
As  we  have  satisfied  equation  (23),  which  results  from  the 

elimination  of  —  —  S—  -^  between  formula  (17)  and  (20),  it  is 
b  r*  sin  £ 

sufficient  to  preserve  one  of  these  last  ;  we  deduce  from  both 


-     =  sn 


The  condition  of  expending  lmc.50  is  expressed  by  formula  (16), 
which  here  becomes 

lmc.50  =  2  *  r'  VW  4/6  g.  0.4226, 
or  else,  by  reducing 

r'VbV  =  0.04785  ......       (a'). 

We  have  still  to  express  the  inequality  (24),  which  gives 

r  >  8  I  sin  45°  or  r  >  5.657  1; 
we  will  take 

r=6b  ........      (a") 

We  have  thus  only  three  equations  involving  b,  V  ,  r,  rf  ;  but 
on  account  of  their  particular  form  we  may  still  deduce  the 
values  of  ft  and  r.  Extracting  the  square  root  of  equation  (a) 
and  multiplying  it  member  by  member  by  (a'),  we  make  r'  V  b' 
disappear  and  find 

b  r  =  0.031107, 
a  relation  which,  combined  with  r  =  6  &,  gives 

r  =  Om.432,        b  =  Om.072. 

This  being  done,  the  system  of  the  three  equations  (a),  (a'),  (a") 
would  no  longer  give  anything  but  r'  VV~;  to  avoid  any  inde- 
termination,  we  will  take  b'  arbitrarily  and  deduce  /•',  except 
that  the  conditions  mentioned  in  (No.  16),  and  not  thus  far 
expressed,  must  subsequently  be  verified.  If  we  take,  for 
example,  b'  =  Om.090,  equation  (ax)  will  become 
/  |/0.090  x  0.072"=  0.04785, 


TURBINES.  95 

whence  we  obtain 

r'  =  Om.594. 

These  values  of  ~b'  and  r'  may  be  retained,  because  —  =  1.37, 

and  the  difference  &'  —  I  —  Om.018  is  only  -  of  r'  —  /•,  a  quan- 

9 

tity  which,  on  account  of  the  obliquity  of  the  floats  to  the 
exterior  circumference,  should  be  but  little  greater  than  two- 
thirds  of  the  length  of  these  last  ;  the  discharge  of  water  will 
not  then  be  too  rapid. 

The  height  h  now  alone  remains  to  be  determined  :  if  the 
level  of  the  tail  race  were  constant,  we  should  make  h  =  o  ; 
however,  we  would  have  to  consider  what  is  said  (No.  16)  on 
this  subject. 

The  theoretical  effective  delivery  will  be  obtained  from 
formula  (21)  ;  we  find 

0.072  0  1  —  cos  25° 


In  practice,  we  only  rely  upon  a  net  effective  delivery  of  from 
0.70  to  0.75  at  most  ;  this  it  is  well  to  do,  on  account  of  all  the 
losses  of  head  that  we  have  neglected,  and  also  because  it  is 
very  difficult  to  make  a  machine  move  exactly  with  the  velo- 
city and  expenditure  of  water  corresponding  to  the  maximum 
effective  delivery. 

Finally,  to  obtain  the  velocity  with  which  the  turbine  should 
revolve,  formula  (14)  should  be  applied,  and  we  would  deduce 

•*  ur 

therefrom  u'  =  10m.555,  since  the  angular  velocity  w  =  —  = 

17.77,  and  finally  the  number  of  revolutions  per  minute  K  = 
?^  =  169.7. 

* 

Again,  let  it  be  proposed  as  a  second  example  to  set  up  one 


96  TURBINES. 

of  Fourneyron's  turbines,  with  a  head  of  2  metres,  expending 
Omc.60  of  water  per  second,  which  corresponds  to  an  absolute 
work  of  1200  kilogrammetres,  or  about  a  16  horse-power.  We 
will  suppose  that  the  water  in  the  tail  race  only  rises  to  the 
level  of  the  lower  plane  of  the  turbine,  so  that  the  interval  or 
play  between  the  turbine  and  the  directing  partitions  may 
communicate  directly  with  the  atmosphere,  and  that  the  pres- 
sure j?  is  sensibly  equal  to  the  atmospheric  pressure.  "We  will 
then  assume  equation  (20  bis),  making  in  it  A"  =  0,  in  other 
words  we  will  place 


Ir1    2  sin  /3  cos 

Following  the  ordinary  rules,  we  will  make  r  —  r'  •  besides,  it 
may  be  remarked  that,  on  account  of  the  position  assigned  to 
the  plane  of  the  tail  race,  h  is  equal  to  h'  with  a  contrary  sign. 
The  above  equation  can  then  be  written 


b    2  sin  |3  cos  j8 

As  &  can  only  differ  slightly  from  90  degrees,  we  will  give  it 
this  value;  the  equation  of  condition  (17)  then  takes  the  form 

I  r*    tanjB  _     ^ 


V  V*    sin  7 

and,  because  r  =  rf 

V  sin  7  =  I  tan  /3 (S) 

Introducing  V  sin  7  in  the  place  of  1)  tan  0  in  the  expression 
(16)  of  the  expenditure,  it  becomes 

Q  =  2  «r  r'  V  sin  7  I^TT. ($") 

The  equations  (£,  <$',  5")  are  those  pertaining  to  the  problem. 
They  contain  six  unknown  quantities,  viz. :  0,  7,  J,  5',  r',  A'  • 
we  consequently  see  that  they  are  indeterminate,  and  that  we 
can  assume  three  of  the  unknown  quantities  or  three  new 
equations.  The  angle  7  not  being  determinable  by  theory,  we 


TURBINES.  97 

will  take  it  at  first  equal  to  30  degrees  (No.  16)  ;  equation  (8") 
will  become,  by  substituting  numbers  for  letters, 

V  r>  =  -54L-  =  0.04312, 
*  V2g 

a  relation  which  is  satisfied  by  the  values 

r'  =  Om.60,         I'  =  Om.072. 

We  thus  see  that  the  ratio  -7  is  only  0.12;  consequently  the 

inequality  in  the  velocities  of  the  liquid  threads  in  the  orifice 
of  exit  will  not  be  too  noticeable.  Now  as  there  remain  three 
unknown  quantities,  J',  k  ',  /3,  connected  only  by  the  two  equa- 
tions (<5),  (<T),  we  will  assume  h'  =  Om.15  ;  then  eliminating 

--  between  (S)  and  ($'),  we  will  obtain 


whence 


and  consequently 

/3  =  42°  40  very  nearly. 
Knowing  £,  we  obtain  from  equation  (<$') 

I  =  Om.039. 

The  difference  1}'  —  1}  =  Om.033  is  perhaps  too  great  relatively 
to  the  height  Om.15  of  the  turbine,  because  the  floats  having  a 
development  of  from  Om.20  to  Om.25  at  the  most,  their  spread 

would  reach  the  amount  of  -  very  nearly.     We  then   try  an- 

other value  of  A'  ;  for  example,  let 

hf  =  Om.30. 
Proceeding  as  before,  we  will  have  successively 

2  cos5  /3  =  ??,  cos  2  /a  =  ^,  ft  =  40°,  ft  =  0-043. 


2  cos2  ft 

"  H  "  40' 

2  cos3  f. 

'  =  -,: 

2  cos2  / 

3  —  1  =  COS  2  jB  JL 

37 

37' 

98  TURBINES. 

The  difference  I'  —  b  would  then  beOm.029;  but  as  the  floats 
would  be  nearly  Om.40  in  length  (on  account  of  their  inclination 
to  the  lower  plane  of  the  turbine),  this  number  appears  per- 
fectly admissible.     We  should  then  obtain  the  results 
7  =  30°,  /s  =  40°,  6  =  90°,  r  =  rf  =  Om.60 
I  =  Om.043,  1'  =  Om.072,  -  h  =  h'  =  Om.30. 
The  effective  delivery  JA  would  be  given  by  equation  (21), 
which,  considering  equation  ($'),  becomes 

p  =  cos  7  =  cos  30°  =  0.866. 
In  like  manner  equation  (14)  would  be  simplified  and  give 

«"  =  ?H, 

whence 

u'  =  4m.429, 

from  this  we  deduce  finally  the  angular  velocity  to  be  given  to 
the  machine 

w  =  —f  =  7.382, 
r 

and  the  number  of  revolutions  per  minute 


In  general,  as  we  have  seen,  the  problem  which  consists  in 
fixing  the  dimensions  of  a  turbine  for  which  we  have  the  ex- 
penditure and  the  head  is  indeterminate  ;  we  take  advantage 
of  this  to  assume  in  part  the  unknown  dimensions,  and  by  the 
method  of  trial,  if  that  be  necessary,  endeavor  to  satisfy  the 
different  conditions  to  be  fulfilled,  but  which  the  equations  do 
not  express. 

18.  Of  the  means  of  regulating  the  expenditure  of  water  in 
turbines.  —  A  turbine  constructed  with  assigned  dimensions,  in 
order  to  move  with  the  maximum  effective  delivery,  should  ex- 
pend a  perfectly  determinate  volume  of  water,  so  long  at  least 
as  the  head  remains  constant.  However,  in  practice,  we  are 


TURBINES.  99 

obliged  to  regulate  the  expenditure  by  the  volume  furnished 
by  the  pond ;  for  if  we  expended  more  we  would  be  exposed  to 
the  lack  of  water,  after  some  time,  and  we  would  then  be 
forced  to  suspend  the  operation  of  the  machine.  Consequently 
we  so  calculate  the  dimensions  as  to  expend  suitably  the  great- 
est volume  of  water  at  our  disposal,  and  we  make  proper  dis- 
positions to  expend  less  when  the  supply  diminishes.  For  this 
purpose  several  methods  have  been  employed. 

In  Fourneyron's  turbine,  the  movable  tub  E  G  F  I  (Fig.  13) 
allows  this  end  to  be  attained :  it  is  merely  necessary  to  lower 
it  more  or  less,  to  contract  the  openings  G  K,  I  L,  or  to  close 
them  entirely.  The  vertical  motion  of  translation  of  this 
tub  is  obtained  by  means  of  three  vertical  rods,  as  r  s,  tu, 
which  are  attached  to  it  at  three  points,  which  form  the  vertices 
of  an  equilateral  horizontal  triangle.  These  rods  are  termina- 
ted at  their  upper  ends  by  screws,  and  enter  into  nuts  which  are 
forced  by  their  construction  to  turn  in  their  places.  The  three 
nuts,  moreover,  are  furnished  with  three  cogged  wheels,  all 
exactly  alike,  which  are  geared  on  the  same  wheel,  which  is 
loose  on  the  axle  of  the  turbine.  By  turning  one  of  the  nuts 
by  means  of  a  wrench,  the  other  two  are  turned  exactly  the 
same  amount,  and  the  tub  is  raised  or  lowered  by  the  three 
rods  at  once. 

There  is  one  great  inconvenience  attendant  upon  the  partial 
obstruction  of  the  openings  G  K,  I  L  ;  it  is  that  the  fluid  veins 
which  issue  through  these  openings  immediately  enter  canals 
of  greater  section,  in  which  they  flow  necessarily  through  a  full 
pipe,  since  the  turbine  is  below  the  tail  race.  There  is  a  sud- 
den change  of  section  thus  produced,  and  consequently  a  great- 
er or  less  loss  of  head.  This  influence  is  sometimes  so  great 
that  General  Morin  has  mentioned,  in  different  experiments  on: 
a  turbine,  a  diminution  in  the  effective  delivery  from  0.79  to- 


100  TURBINES. 

0.24,  when  the  free  opening  under  the  wheel  descended  from 
its  greatest  elevation  to  about  ^  of  this  height.  The  inconve- 
nience is  so  much  the  greater  as  the  diminution  in  the  effective 
delivery  corresponds  to  that  of  the  volume  of  water  expended, 
which  tends  to  make  the  dynamic  effect  of  the  machine  very 
irregular.  To  remedy  this,  M.  Fourneyron  has  proposed  to 
subdivide  the  height  of  the  turbine  into  several  stages,  by  means 
of  two  or  three  annular  horizontal  plates,  like  the  plates 
S  R  M  N,  TJ  T  P  Q,  the  distance  between  which  is  divided  into 
three  or  four  equal  parts.  Supposing,  for  example,  that  there 
are  three  stages,  we  see  that  there  will  be  no  sudden  change  of 
section  when  the  cylindrical  sluice  is  raised  J,  J,  or  the  whole 
height  of  the  turbine  ;  in  all  cases  the  phenomenon  of  sudden 
expansion  will  only  affect  a  portion  of  the  liquid  vein.  But, 
on  the  other  hand,  the  construction  of  the  machinery  is  compli- 
cated, and  the  friction  of  the  water  against  the  solid  walls  is 
increased.  M.  Fourneyron  has  again  proposed  using  only  the 
two  plates  S  E-  M  N,  U  T  P  Q  ;  the  lower  one  would  carry  the 
floats  only,  and  the  upper  one  be  pierced  with  grooves  which 
would  allow  it  to  settle  freely  between  the  floats,  under  the 
action  of  its  weight  alone.  This  upper  plate  would  bear  on  a 
flange  placed  at  the  bottom  of  the  cylindrical  sluice  on  the  out- 
side. When  the  sluice  is  lowered,  the  plate  S  R  M  N  is  low- 
ered with,  it,  and  the  height  of  the  turbine  would  be  always 
equal  to  the  height  to  which  the  sluice  is  raised.  The  movable 
plate  carried  around  by  the  turbine  would,  while  turning,  rub 
against  the  flange  which  serves  it  as  a  support ;  but,  the  pres- 
sure between  the  two  being  slight,  this  would  not  produce  an 
increase  of  resistance  worthy  of  mention. 

M.  Fontaine,  to  regulate  the  expenditure  of  his  turbines, 
uses  a  series  of  valves  similar  to  that  represented  in  Fig.  17. 
A  B  is  a  directing  partition,  B  C  a  float  of  the  turbine,  D  a 


TURBINES. 


101 


valve  which  can  be  sunk  to  a  great  or  less  depth  into  the  space 
between  A  B  and  the  next  directing  partition  to  the  left.  In 
this  way  we  can  narrow  as  much  as  we  choose  the  free  pas- 
sage into  this  interval,  and  as  the  same  effect  is  produced  on  all 
in  the  same  way,  it  is  plain  that  we  have  the  means  of  reducing 
the  volume  of  water  expended  as  much  as  circumstances  may 
require.  The  vertical  motion  of  translation  is  given  simultane- 
ously to  all  the  rods  E  F  by  means  similar  to  those  used  by 
M.  Fourneyron  :  these  rods  are  all 
connected  with  a  metallic  ring, 
at  three  points  of  which  are  at- 
tached vertical  screws,  furnished 
with  nuts  which  can  be  turned 
only  in  their  beds.  These  three 
have  each  a  cogged  wheel  secure- 
ly attached,  the  wheels  being  all 
of  the  same  size,  surrounded  by 
an  endless  chain,  which  causes 
them  to  turn  equally  and  at  the 
same  time.  It  is  then  necessary  to  turn  one  of  the  three 
wheels  by  means  of  a  winch  and  pinion,  in  order  that  the  whole 
system  of  valves  may  take  up  a  vertical  motion.  The  partial 
closing  of  the  in-leading  canals  here,  as  in  Fourneyron's  tur- 
bine, is  not  without  inconveniences,  for  the  sudden  change  of 
section  in  these  canals  still  gives  rise  to  a  loss  of  head ;  how- 
ever, this  loss  is  found  to  be  diminished  to  a  considerable 
extent. 

For  the  valves  M.  Koecklin  has  substituted  clack-valves  re- 
volving on  a  hinge,  so  as  to  fit  exactly  over  the  entrance  to  the 
distributing  canals,  when  the  closing  is  complete.  The  ar- 
rangement of  the  Koecklin  turbine  also  allows  the  expenditure 
of  water  to  be  regulated  by  making  use  of  the  sluice  Y  (Fig. 


FIG.  17. 


102  TURBINES. 

15),  which  can  close  the  communication  between  the  well 
placed  below  the  turbine  and  the  tail  race.  But  experience 
shows  that  a  greater  portion  of  the  head  is  lost  in  this  way  than 
by  using  the  valves. 

The  inconveniences  attendant  upon  the  partial  closing  of  the 
distributing  channels  are  so  great,  when  considered  from  the 
point  of  view  of  economy  of  the  motive  power,  that  all  possible 
means  for  remedying  them  have  been  sought  for.  We  have 
already  mentioned  two  devised  by  M.  Fourneyron.  M.  Charles 
Gallon,  civil  engineer,  and  a  constructor  of  reputation,  has  pro- 
posed another  way,  which  consists  in  making  all  the  partial 
sluices,  which  close  the  channels  in  question,  independent  of 
each  other ;  to  diminish  the  expenditure,  a  certain  number  of 
these  sluices  can  be  closed  completely,  leaving  the  remainder 
entirely  opened.  But  as  the  channels  formed  by  the  floats  of 
the  turbine  pass  alternately  before  the  opened  and  closed  ori- 
fices, there  is  still  a  cause  of  unsteadiness  and  trouble  in  the 
motion. 

M.  Gallon's  idea  has  been  reproduced  under  another  form  by 
M.  Fontaine.  The  orifices  of  entrance  of  the  distributing  chan- 
nels occupy  a  horizontal  surface  comprised  between  two  circles 
concentric  with  the  axis  of  the  turbine,  M.  Fontaine  arranges 
two  rollers,  of  the  shape  of  a  truncated  cone,  which  can  roll 
over  this  annular  surface.  The  two  rollers  are  mounted  on  the 
same  horizontal  spindle  which  forms  a  collar  surrounding  the 
axle  of  rotation.  When  they  move  in  one  direction,  each  one 
of  them  unrolls  a  band  of  leather,  which  has  one  end  fastened 
to  the  roller  and  the  other  to  the  plane  of  the  orifices  of  en- 
trance :  some  of  the  orifices  are  thus  entirely  closed  while  the 
others  remain  wide  open.  When  the  truncated  cones  move  in 
the  contrary  direction,  they  roll  up  the  two  leathern  bands  and 
uncover  the  orifices.  M.  Fontaine  has  also  imitated  the  tur- 


TURBINES.  103 

bines  of  several  stories  of  M.  Fourneyron,  in  proposing  to  di- 
vide the  turbines  into  several  zones  by  means  of  surfaces  of 
revolution  about  the  axis  of  the  system ;  each  of  these  zones 
could  be  separately  obstructed. 

19.  Hydropneumatic  turbine  of  Girard  and  Gallon. — The 
problem  of  regulating  the  expenditure,  without  too  great  loss, 
appears  to  have  been  solved  in  the  best  manner  in  a  kind  of 
turbine  called  by  its  inventors,  MM.  Girard  and  Gallon,  the 
hydropneumatic  turbine.  Their  system  consists  essentially  in 
surrounding  Fourneyron's  turbine  with  a  sheet-iron  bell,  the 
lower  plane  of  which  is  nearly  at  the  height  of  the  points  at 
which  the  water  leaves  the  floats.  In  this  bell,  by  means  of  a 
small  pump  set  in  motion  by  the  machine  itself,  the  air  is  com- 
pressed, which  gradually  forces  the  water  entirely  out  of  the 
bell ;  then,  if  we  suppose  the  cylindrical  sluice  to  be  partially 
raised,  the  liquid  vein  which  escapes  below  has  a  depth  less 
than  the  distance  between  the  two  plates  of  the  turbine ;  but' 
no  sudden  change  in  the  section  of  liquid  results  from  this,  be- 
cause the  turbine  moves  in  compressed  air,  and  it  is  not  cover- 
ed by  the  water  in  the  tail  race.  The  water  flowing  into  the 
turbine  has  a  depth  which,  at  the  beginning  of  the  floats,  is 
equal  to  the  height  to  which  the  sluice  is  raised ;  the  upper 
plate  is  no  longer  wet,  and  as  it  only  serves  to  hold  the  floats, 
it  can  be  hollowed  out,  in  order  to  make  sure  of  a  free  circula- 
tion of  air  above  the  liquid  vein.  Thus  the  principal  cause  of 
the  loss  of  head,  due  to  the  partial  raising  of  the  sluice,  is 
found  to  be  suppressed,  and  we  should  succeed  in  obtaining  a 
very  slightly  varying  effective  delivery. 

The  calculations  to  be  employed  for  the  hydropneumatic  tur- 
bine may  be  regarded  as  a  particular  case  of  those  in  No.  15. 
Preserving  the  same  notation,  we  must  consider  bf  as  an  un- 
known quantity,  and  at  the  same  time  suppose 


104-  TURBINES. 

P—p'-^Pa  +  n^; 

in  fact,  h  represents  the  depth  to  which  the  turbine  is  immersed 
below  the  level  of  the  tail  race,  and  pa  -f  n  h  is  properly  the 
pressure  of  the  air  in  the  bell.  Thus,  from  this  value  of  p  equa- 
tions (19)  and  (20)  show  first  that  k  —  1,  and  consequently  (No. 
16)  that  we  have  2  ft  +  6  =  180°,  the  only  condition  to  be  satis- 
fied by  the  dimensions  of  the  machine.  It  gives  ft  +  d  =  180° 
-  ft  ;  equation  (17)  then  becomes 

~b'  r'*  sin  7  =  b  r*  sin  6  —  b  r*  sin  2  ft  ; 

whence  we  can  deduce  V  in  terms  of  the  height  I  to  which  the 
sluice  is  raised,  for  a  turbine  moving  with  the  maximum  effec- 
tive delivery.  In  virtue  of  the  preceding  relation,  equations 
(13),  (16),  and  (21)  take  the  form 


, 

sin/       2cos-/r 


Q  =  2  *  5  r  sin  /3  V  2  g  H, 

r'* 

f*  =  1  —  TT— ^ j—  (1  —  cos  7) ; 

2  r2  cos  ft 

these  formulas,  which  can  very  readily  be  demonstrated  direct- 
ly, give  :  1st,  the  angular  velocity  -  of  the  turbine,  which  an- 
swers to  the  maximum  effective  delivery;  2d,  its  expenditure 
Q,  and  its  effective  delivery  ^  under  the  same  condition.  If  we 
had  to  set  up  a  turbine  for  a  given  head  and  expenditure,  the 
equation  of  which  Q  is  the  first  member,  together  with  the  in- 
equality (24)  (No.  16),  would  give  us  the  means  for  finding  r 
and  b  sin  /3,  and  consequently  &,  after  ft  has  been  chosen.  As 
to  7,  it  should  always  be  taken  between  20  and  30  degrees ;  6 
should  be  equal  to  180°  —  2/3;  finally,  r'  should  be  taken  as 
small  as  possible  (on  account  of  the  expression  for  (a),  but  not, 
however,  so  as  to  run  any  risk  of  making  the  floats  too  short. 

The  method  of  MM.  Girard  and  Gallon  could  be  equally  well 
adapted  to  Fontaine's  turbine*. 


TURBINES.  105 

20.  Some  practical  views  on  the  subject  of  turbines. — The 
directing  partitions  and  floats  are  generally  made  of  sheet  iron  ; 
they  are  fastened  to  the  surfaces  that  are  to  support  them  either 
by  angle  irons,  or  by  setting  them  in  cast-iron  grooves  on  these 
surfaces.  They  should  be  sufficient  in  number  to  give  to  the 
velocity  of  the  water  their  own  direction.  The  distance  of  any 
two  consecutive  floats  or  partitions  apart  should  not  be,  at  any 
point,  more  than  Om.06  to  Om.08,  measured  along  the  normal  to 
the  surfaces,  and  generally  it  is  made  less.  However,  it  must 
not  be  made  too  small,  for  then  the  friction  of  the  water  against 
the  solid  sides  would  be  too  great. 

As  the  floats  in  Fourneyron's  turbine  are  placed  further  from 
the  axis  than  the  partitions,  that  is  to  say  distributed  over  a 
greater  circumference,  their  number  is  from  one-third  to  one- 
half  greater  than  that  of  the  partitions,  in  order  to  have  every- 
where a  suitable  distance  apart. 

Excepting  the  condition  of  cutting  the  planes  of  the  orifices 
under  a  determinate  angle,  the  curvature  of  the  floats  and  par- 
titions is  almost  a  matter  of  indifference.  However,  as  too 
great  a  curvature,  or  a  sudden  change  in  the  curve,  may  pre- 
vent the  threads  from  following  the  sides,  and  thus  produce 
losses  of  head,  we  must  avoid  these  two  faults.  It  would  be 
well  to  have  the  radius  of  curvature  at  least  three  or  four  times 
the  distance  apart  measured  along  the  normal. 

Turbines  can  be  used  for  all  heads  and  every  expenditure. 
For  example,  some  are  mentioned  the  head  being  only  from  Om.30 
to  Om.40,  whilst,  in  the  Black  Forest,  there  is  a  turbine  set  up  by 
M.  Fourneyron  which  has  a  fall  of  108  metres.  The  expendi- 
ture may  be  great  even  with  quite  small  dimensions.  In  one 
of  the  examples  given  in  No.  17,  we  have  seen  that  a  turbine 
of  Om.60  external  radius  and  Om.09  in  height,  expended  without 
loss  lmc.50,  or  1500  litres  per  second.  Some  turbines  are  con- 


106  TTJKBINES. 

i 

structed  whose  expenditure  reaches  as  high  as  4  cubic  metres, 
or  4000  litres,  per  second,  and,  if  need  be,  more  could  be  ex- 
pended. 

Under  ordinary  circumstances,  turbines  move  with  sufficient 
rapidity,  and  thus  allow  the  gearing  for  transmission  to  be 
economized. 

For  each  turbine  organized  with  fixed  conditions,  the  theory 
of  'No.  15  gives  a  certain  velocity  to  be  imparted  to  it,  in  order 
to  obtain  the  maximum  useful  effect.  But  if,  in  reality,  a  velo- 
city different  from  this  be  given  to  it,  and  deviate  from  25  per 
cent,  more  or  less,  experience  shows  that  the  effective  delivery 
does  not  change  much,  a  very  important  property  for  many 
manufacturing  purposes,  in  which,  in  spite  of  the  variations 
that  take  place  in  the  expenditure  of  the  head  of  water,  it  is 
important  to  have  the  machine  always  move  with  a  very  nearly 
constant  velocity. 

These  motors  have  then  a  very  decided  advantage  over 
wheels  with  a  horizontal  axis.  Unfortunately  their  propor- 
tional effective  delivery  is  not  always  constant,  even  approxi- 
mately, for  heads  with  a  very  variable  expenditure ;  their  con- 
struction and  repairs  can  only  be  intrusted  to  very  skilful  me- 
chanics, and  consequently  are  quite  expensive ;  while,  on  the 
other  hand,  overshot  and  breast  wheels  can  be  very  economi- 
cally constructed,  and  at  the  same  time  give  an  effective  delivery 
at  least  equal  to,  if  not  greater  than,  that  of  turbines.  These 
wheels  will  frequently,  on  this  account,  be  preferred  when  the 
expenditure  and  head  of  water  are  favorable  to  their  con- 
struction. 


KEAOTION  WHEELS. 


21.  Reaction  wheels. — Let  us  conceive  of  one  of  Fourney- 
ron's  turbines  deprived  of  its  directing  partitions,  care  being 
taken  to  prolong  the  floats  to  a  slight  distance  from  the  axis  of 
rotation ;  let  us  suppose  that  the  water  comes  to  the  floats 
through  a  pipe  concentric  with  the  axle,  having  for  a  radius 
precisely  the  free  distance  that  we  have  just  mentioned.  Fur- 
thermore, the  expenditure  of  water  in  this  pipe  will  be  consid- 
ered as  very  little/  in  order  that  the  absolute  velocity  of  the 
liquid  in  it  may  not  be  sensible.  We  shall  thus  have  the  idea 
of  reaction  wheels. 

To  give  their  theory,  we  will  consider  the  point  of  entrance 
of  water  into  the  wheel  as  being  on  the  axis  of  rotation  itself; 
at  this  point  the  velocity  of  the  wheel  being  nothing,  as  well 
as  the  absolute  velocity  of  the  water,  it  will  be  the  same  for 
the  relative  velocity  of  this  last.  In  the  calculations  of  ~No. 
15,  we  will  have  to  make  v  =  o,  u  =  o,  w  =  0y  equations  (1), 
(2),  and  (3)  then  become 

o  =  H  +  A  -h  Pa~P 
n 

P—l  ' 


whence  we  find 

w'"  =  2  g  H  +  uf\ 


108  REACTION   WHEELS. 

.Neglecting  friction,  the  only  loss  of  head  is  as  yet  the  height 

vf* 

—  due  to  the  absolute  velocity  of  exit ;  we  calculate  its  value 

?9 

by  means  of  equation  (6),  which,  combined  with  the  preceding, 
will  give 

•v"  =  2  w/a  +  2  g  H  —  2  u'  cos  7  V  2  g  H  +  w*. 
The  effective  delivery  would  then  be 

H— £ 

2  g 

*  =  —-'  =  l  " 


20! 

x 


,   . 

or,  placing 

p  =  —  x*  +  x  cos  7  \/  2  +  #2. 

We  can  consider  /x  as  a  function  of  a?,  and  seek  its  maximum 
when  x  varies.  To  this  end  we  will  take  off  the  radical  sign 
by  writing 

(A*  +  #3)a  =  x*  cos2  7  (2  +  #'), 
or  by  reducing 

a?4  sin3  7  —  2  a?3  (cos2  7  —  M-)  +  ^  =  0. 

If  we  deduced  from  this  a?2  in  terms  of  f*,  the  roots  would 
become,  from  their  nature,  real;  consequently,  we  have  the 
condition 

(cos2  7  —  (x)2  —  fx2  sin2  7  >  0, 

or  developing  and  reducing 

cos4  7  —  2  f*  cos2  7  +  M-2  cos2  7  >  0. 
If  we  suppress  the  positive  factor  cos2  7,  we  find 

cos2  7  -  2f*  +  M-2  >  0, 
or 

(1  -  M-)'  _  sin2  7  >  0, 


REACTION   WHEELS. 


109 


and,  observing  that  1  —  M-  is  necessarily  p-jsithe,  ao  wc.l  as 
sin  7, 

1  —  fx  >  sin  7, 
(*  <  1  —  sin  7. 
The  limit  of  the  effective  delivery  that  we  can  reach  is  then 

fXj  =  1  —  sin  7. 

The  corresponding  value  for  x  is  easily  obtained  by  the  relation 
between  x  and  f*  ;  if  we  take  the  equation  deprived  of  radicals, 
and  make  in  it  x  =  x^  and  ji.  =  1  —  sin  7,  it  becomes 

os*  sin2  7  —  2  x?  sin  7  (1  —  sin  7)  +  (1  —  sin  7)"  =  0, 
or,  more  simply,  by  extracting  the  square  root 

x?  sin  7  —  (1  —  sin  7)  =  0, 
whence 


/ 

=  \/ 
V 


1  —  sin  7 

sin  7 


If  we  supposed  7  =  0,  we  would  find  f*l  =  1  ;  but  xl9  and 
consequently  w',  would  become  infinite.  Strictly  speaking,  the 
value  7  =  o  might  be  realized  ;  it  would  only  be  necessary 
that  the  channels  that  make 
up  the  wheel  should  be  ar- 
ranged no  longer  side  by 
side  without  empty  spaces, 
as  in  turbines,  but  according 
to  the  annexed  sketch  (Fig. 
18).  We  should  have  a 
certain  number  of  curved 
plates,  as  AB,  meeting  the 
circumference  OB  at  B, 
where  they  end,  and  com- 
municating  at  A  with  the  supply-pipe,  to  which  they  are  per- 
manently attached.  The  supply-pipe  would  then  form  the 
axle  of  rotation.  But,  in  adopting  this  arrangement,  u'  could 


FlG-  18« 


110  REACTION   WHEELS. 

not  become  infinite,  nor  j*,  consequently,  reach  unity :  we  only 
see  that  it  would  be  necessary  to  make  the  wheel  turn  very 
rapidly.  Moreover,  it  is  difficult  to  expend  a  large  volume  of 
water  without  giving  a  considerable  diameter  to  the  central 
pipe  A  O,  which  would  make  the  hypothesis,  that  u,  v,  and  w 
were  equal  to  zero,  inadmissible ;  there  are  equally  great  diffi- 
culties in  regulating  the  expense  according  to  the  required  cir- 
cumstances. It  is  undoubtedly  on  these  accounts  that  this  kind 
of  wheel  is  but  little  used. 

preserving  the  arrangement  of  the  floats  of  Fourneyron's 
turbine,  which  gives  a  series  of  contiguous  channels,  we  can  no 
longer  make  7  —  0,  and  then  the  limit  of  the  theoretical  effec- 
tive delivery  decreases  quite  rapidly  as  7  increases ;  so  that,  for 
7  =  15°,  1  —  sin  7  would  no  longer  be  greater  than  0.741.  On 
the  other  hand,  as  we  offer  the  water  a  wider  outlet,  we  should 
perhaps  lose  less  in  friction,  and  the  theoretical  effective  deliv- 
ery would  differ  less  from  the  real. 


PUMPS. 


§  IV.    OF  A  FEW  MACHINES  FOR  RAISING  WATER. 

22.  Pumps. — The  arrangement  and  shape  of  the  parts  of 
pumps  are  of  infinite  variety,  according  to  the  notions  of  the 
constructor.  A  special  treatise  would  be  necessary  merely  to 
describe  the  principal  kinds.  We  will  suppose,  then,  that  the 
reader  Las  seen  a  summary  description  of  these  machines,  and 
confine  ourselves  to  general  ideas. 

(a.)  Effort  necessary  to  make  the  piston  move. — Two  cases 
must  be  distinguished :  pumps  of  single  stroke,  and  pumps  of 
double  stroke.  In  the  former  case,  the  piston  only  draws  up 
the  water  into  the  pump,  or  else  only  drives  out  the  water  pre- 
viously drawn  up,  by  forcing  it  up  through  the  delivery  pipe, 
when  it  moves  in  a  determined  direction ;  in  the  second,  these 
effects  are  produced  simultaneously,  whatever  be  the  direction 
of  the  stroke.  Let  us  first  take  the  single  stroke  ;  let 

h  be  its  height ; 

&  the  section  of  the  piston  ; 

pa  the  atmospheric  pressure  ; 

n  the  weight  of  a  cubic  metre  of  water. 

The  side  of  the  piston  in  contact  with  the  column  of  water  drawn 
up  would  support,  supposing  that  it  remained  in  equilibrio,  a 
pressure  equal  to  &  (pa  —  n  A),  whilst  the  other  side,  generally  in 
direct  communication  with  the  atmosphere,  would  sustain  a 
pressure  in  a  contrary  direction  equal  to  pa  &  ;  the  difference 


112  PUMPS. 

n  n  A  would  be  the  resultant  pressure  on  the  piston.  If,  on 
the  contrary,  there  be  but  a  single  stroke  forcing  to  a  height  A', 
we  will  find,  in  like  manner,  that  the  piston  supports,  exclu- 
sively of  its  motion,  a  resultant  pressure  n  n  A'.  Finally,  if 
the  pump, were  one  of  double  stroke,  these  two  resultants  would 
be  superposed,  and  the  value  for  the  total  pressure  would  be 
n  ft  (A  -f  A')  or  n  a  H,  H  being  the  height  included  between 
the  level  of  the  basin  from  which  the  water  is  drawn,  and  that 
of  the  basin  receiving  it.  In  a  certain  class  of  single-stroke 
pumps,  the  same  superposition  of  resultant  pressure  on  the  two 
faces  of  the  pis.ton  takes  place  during  the  motion  in  one  direc- 
tion, and  these  resultants  are  in  equilibrio  when  motion  occurs 
in  the  opposite  direction :  this  is  the  case  in  the  kind  of  pump 
called  the  lifting-pump.  If  the  piston  have  no  horizontal 
motion,  we  must,  besides,  consider  its  weight  as  well  as  that  of 
its  rod,  the  component  of  which,  parallel  to  the  axis  of  the 
pump,  would  be  added  to  or  subtracted  from  the  preceding 
expressions,  according  to  circumstances.  The  friction  of  the 
piston  against  the  barrel  of  the  pump  must  also  be  added,  as 
well  as  that  of  the  rod  against  the  packing-box,  if  there  be  any, 
through  which  it  passes. 

But  these  expressions  only  give  the  value  of  the  force  capable 
of  maintaining  the  piston,  as  well  as  the  water  drawn  or  forced 
up,  in  equilibrio  in  a  given  position.  When  motion  takes 
place,  the  effort  brought  to  bear  on  the  piston  may  be  very  dif- 
ferent from  this  force.  In  the  first  place,  the  water  does  not 
begin  to  move  in  the  pipes,  and  does  not  pass  through  the  nar- 
row openings  of  the  valves  without  experiencing  losses  of  head 
which  are  to  be  added  to  the  heights  h  and  h'.  For  ex- 
ample, in  the  case  of  the  sucking  pump,  if  there  be  a  loss  of 
head  £  in  the  length  of  the  column  drawn  up,  the  pressure 
n  (pa  —  n  A)  will  be  reduced  to  ii  (pa  —  n  (A  -f  f) ),  and  the 


PTJMPS.  113 

resultant  n  n  h  would  become  n  a  (A  -f  £).  In  like  manner, 
if  we  consider  a  single  forcing  stroke,  and  that  there  is  in  the 
entire  column  forced  up  a  loss  of  head  £',  the  piezometric  level 
in  this  column,  at  the  point  at  which  it  touches  the  piston, 
would  be  increased  by  £',  and  the  resultant  pressure  would  be 
n  n  (hf  4-  £7).  If  the  pump  be  one  of  double  stroke,  the  ex- 
pression n  n  H  should  in  like  manner  be  replaced  by  n  n  (H 
+  ?  H~  £0-  Besides  the  heights  £  and  £',  others  must  be  added, 
if  the  mass  of  water  set  in  motion  is  not  uniformly  displaced. 
Let  P  be  the  weight  of  the  piston  and  its  rod?t;  its  acceleration, 
P'  the  weight  of  the  water  set  in  motion  and  which  fills  the 
pipes  through  which  it  is  drawn  up  or  forced  out :  the  weight 
P'  being  supposed  to  move  with  a  mean  acceleration  j',  we  see 

that  an  additional  force  -(Pj  4-  ¥' jf)  would  be  necessary  to 
y 

overcome  the  inertia  of  the  water  and  piston.  This  additional 
force,  at  one  time  a  pressure,  at  another  a  resistance,  may  pro- 
duce considerable  variations  in  the  total  force  to  be  applied  to 
the  piston,  which  is  always  inconvenient :  since  we  must,  in 
the  first  place,  determine  the  dimensions  of  the  pieces,  not  ac- 
cording to  the  mean,  but  according  to  the  maximum  effort  sus- 
tained, which  generally  produces  a  clumsy  and  expensive 
machine ;  in  the  second,  it  is  seldom  that  great  variations  in 
resistance  do  not  give  rise  indirectly  to  some  loss  of  motive 

Pf 

power.     We  diminish  the  value  of  the  term  — -  by  means  of 

y 
weights  acting  as  a  counterpoise  to  the  piston,  if  P  is  large ;  f 

P/    n1 

and  consequently — ^-  are  also  diminished,  in  case  it  is  found 

9 
to  be  worth  the  trouble,  by  means  of  an  air-chamber,  placed 

at  the  entrance  to  the  ascent  pipe,  which  makes  the  motion  of 
a  great  part  of  the  weight  of  the  column  forced  up  uniform. 


114 


PUMPS. 


and    consequently  suppresses  or  diminishes    to  a  very  great 
extent  the  corresponding  force  of  inertia. 

Another  means  of  obtaining  approximate  uniformity  of  mo- 
tion in  the  ascent  pipe  consists  in  making  it  answer  for  the 
delivery  of  several  pumps  working  together,  in  such  a  way 
that  their  total  delivery  in.  a  series  of  equal 
times  shall  vary  but  little :  it  is  accomplished 
/  \  in  this  way.  Let  O  (Fig.  19)  be  the  axis  of 
rotation  of  an  arbor  that  receives  from  a 
motor  a  motion  rendered  regular  by  a  fly- 
wheel, and  consequently  nearly  uniform. 
This  arbor  carries  two  arms,  O  B,  O  B',  mak- 
ing a  right  angle  with  each  other ;  to  each 
of  them  is  attached  a  connecting  rod,  fasten- 
ed at  its  other  end  to  a  piston  running  be- 
tween guides,  and  which  belongs  to  a  double-stroke  pump.  As 
an  example,  we  will  suppose  the  arbor  O  horizontal,  the  piston 
rods  vertical  and  their  prolongations  intersecting  the  axis  of 
rotation ;  the  connecting  rods  will  generally  be  of  the  same 
length,  about  five  or  six  times  as  long  as  the  arm  O  B.  It  fol- 
lows from  this  that  the  obliquity  of  the  connecting  rods  with 
the  vertical  being  always  small,  the  velocities  v  and  v'  of  the 
pistons  are  practically  those  of  the  projections  of  B  and  B'  on 
the  vertical  B0  B: ;  calling  w  the  angular  velocity  of  the  arbor, 
J  the  length  O  B,  a?  the  angle  formed  by  O  B  with  the  vertical 
B0  Biy  we  will  then  have 


v  =  w  1}  sin  a?,  vf  =  u  ~b  sin  ( -   -f  x  )  =  w  5  cos  x. 

\A 

Let  n  again  be  the  common  cross-section  of  the  two  pistons, 
and  &  a  very  short  interval  of  time ;  exclusively  of  the  losses 
by  the  play  of  the  two  mechanisms,  the  volume  of  water  fur- 
nished during  the  time  &  to  a  common  ascent  pipe,  by  the  two 


PUMPS.  115 

pumps  together,  will  be  the  arithmetical  sum  of  the  volumes 
generated  by  the  two  pistons,  viz.,  Q  (v  +  v')6  or  n  w  b&  (sin  x 
+  cos  #),  a  formula  in  which  the  sine  and  cosine  should  have 
their  absolute  values  given,  since  it  is  a  question  of  an  arith- 
metical sum,  and  therefore  the  velocities  are  essentially  positive. 
Consequently  it  is  sufficient,  in  order  to  obtain  the  maximum, 
the  minimum,  and  the  mean  of  the  variable  quantity  sin  x  + 

cos  a?,  to  suppose  x  included  between  o  and  -.    Now  we  find  be- 

2 

tween  these  limits 

Two  minima  equal  to  1  for  x  =  o  and  x  =  ~ 

One  maximum  equal  to  1.414,  for  x  =  j  j 

if 

rz 

mi                   i              i  ,     «/ «    (sin  x  4-  cos  a?)  d  x       4:       .,  nfrc. 
The  mean  value  equal  to  — — '- — '- —  =  -  =  1.272. 

if  * 


'2 
0    dx 

There  would  thus  be  between  the  minimum  and  the  mean  a 

0  272 
relative  difference  of  7^9  >  or  a^out  0.214  ;  whilst,  with  a  sin- 


S: 


gle  pump,  the  elementary  delivery,  proportional  to  sin  a?,  would 
vary  between  0  and  1,  and  would  have  -  or  0.637  for  a  mean 

value,  which  would  produce  a  much  greater  relative  difference 
between  the  minimum  and  the  mean. 

We  obtain  a  still  more  satisfactory  result  when  we  use  three 
arms  making  angles  of  120°  with  each  other.  The  elementary 
delivery  of  the  three  pumps  together  is  then  proportional  to 

/         2tf\  /         4r#\ 

sin  x  +  sin  (x  +  —-J  +  sm  \x  4-  -^  J, 

each  sine  to  be  always  taken  as  positive,  whatever  may  be  x. 
8 


116  PUMPS. 

We  readily  see,  moreover,  that  the  arithmetical  sum   of  the 
three  sines  will  not  change  by  increasing  the  arc  by  60  degrees, 

so  that  it  is  sufficient  to  make  x  vary  between  0  and  -.   Within 

o 

these  limits,  the  first  two  sines  are  positive  and  the  third  nega- 
tive ;  hence,  the  sum  of  the  absolute  values  will  be  expressed 


sm  x  +  sin  (x  +  y )  —  sin  (x  4-'-^), 

( 

».s  ' 

3 


or  else,  by  developing  and  observing  that  the  arcs  —  and  — 


together  make  an  entire  circumference, 
or  finally 


sin  a?  -f  2  cos  x  sin  --- 
o 


sin  x  +   V  3  cos  #. 
The  minima  of  this  quantity  correspond  to  x  =  0  and  x  =    , 

and  have  for  their  value  V1T  or  1.732 ;   the  maximum,  corre- 
sponding to  x  =  -,  is  -  +  -  V~3.  V~3,  that  is  2 ;   the  mean 
if 

rl  ._ 

-M^.Wrcos^*  becomeg  _6  or  1  _910_     The  relatiye 


dx 
difference  between  the  minimum  and  the  mean  is  consequently 

I  910 i  Y32 

lowered  to  -   --T-QTVT — -  or  to  about  0.093. 

There  is  also  a  great  deal  of  regularity  in  the  elementary 
delivery  of  the  three  pumps  united  as  above,  when  they  are 
supposed  to  be  of  a  single  stroke.  Let  us  admit,  for  example, 


PUMPS.  117 

that  each  piston  only  forces  up  water  when  its  crank  O  B  de- 
scends from  B0  to  Bx  ;  the  sum  of  the  elementary  deliveries  will 
still  be  proportional  to  the  expression 

/  2  tf\  /  4:  lJf\ 

sin  x  +  sm(  x  -f  —  J  +  sin  (x  +  —  J  ; 

but  the  pumps  being  only  of  single  stroke,  instead  of  changing 
the  sign  of  the  negative  sines,  we  must  suppress  them  altogether. 

This  granted,  let  us  first  increase  x  from  0  to  ~  :   #  and  x  -4- 

o 

—  will  be  less  than  the  semi-circumference,  and  x  H  —  -  will  be 
3  3 

included  between  if  and  2  *.     "We  shall  then  only  have  to  keep 

within  these  limits  the  sum  sin  x  -f  sin  (x  H  —  -\  which  can 

\          3  / 

be  put  under  the  form 

C°S      °r  S 


since  cos  o  =  „  J  ^n^s  sum?  equal  to  sin  „,  or  0.866  for  x  =  0, 
becomes  a  maximum  and  equal  to  1  for  a?  =  ^,  then  it  decreases 

to  0.866  when  x  passes  from  ^  to  -.     In  the  second  place,  if  we 

tf         9  <jf  2  if 

take  the  values  of  x  between  -  and  —  ,  the  sines  of  x  +  —  -  and 

o  o  o 

x  -f  —  are  both  negative,  so  that  the  sin  x  alone  can  be  pre- 

served, which  has  still  a  minimum  value  0.866,  corresponding 
to  two  limits,  and  1  for  a  maximum  equidistant  from  these 
limits.  Moreover,  it  is  useless  to  consider  the  values  of  x 

greater  than  —  ,  for  a  rotation  of  120  degrees  not  producing 
3 

any  change  of  figure  in  the  combination  of  the  mechanism,  we 


118  PUMPS. 

would  again  find  the  same  sines.  We  see,  then,  that  the  ele- 
mentary delivery  of  the  three  pumps  working  together  varies 
as  certain  numbers  which  are  always  comprised  between  0.866 
and  1,  and  consequently  that  it  is  sufficiently  regular:  the 
minimum  and  maximum  are  respectively  half  of  what  they 
were  in  the  case  of  three  pumps  of  double  stroke. 

We  have  heretofore  supposed  that  the  two  cranks  at  right 
angles  to  each  other,  or  the  three  cranks  following  each  other 
at  distances  of  120  degrees,  are  fixed  to  the  same  arbor ;  it  is 
plain  that  they  can  be  attached  to  different  arbors,  provided 
that  they  all  have  the  same  angular  velocity,  the  cranks  being 
of  equal  length ;  or,  more  generally,  provided  that  the  centre 
of  the  joint  of  each  of  them  with  the  corresponding  connecting 
rod,  in  the  three  systems,  has  the  same  velocity  of  rotation 
about  its  arbor. 

It  is  always  well,  as  has  been  already  said,  to  avoid  great 
variations  in  the  force  that  is  to  be  transmitted  to  the  piston  of 
a  pump ;  this  becomes  almost  indispensable  when  it  is  moved 
by  means  of  horses.  An  essential  condition  for  employing  the 
work  of  horses  to  advantage  is  that  their  pace  and  the  exertion 
they  have  to  make  shall  be  as  steady  as  possible  ;  it  would  be 
difficult  to  accomplish  this  with  one  single  stroke  pump  forcing 
up  a  long  column  of  water,  the  piston  of  which  should  receive 
its  motion  from  the  arbor  of  the  gearing-wheel  by  a  system  of 
a  crank  and  connecting  rod,  and  we  could  only  succeed  by  em- 
ploying fly-wheels,  more  or  less  clumsy.  It  would  generally 
be  better  to  regulate  the  resistance  by  means  now  to  be  men- 
tioned. 

(J)  Work  to  be  transmitted  to  the  piston. — If  we  knew  exact- 
ly, in  each  position  of  the  piston,  the  force  to  be  applied  to  give 
it  motion,  it  would  be  easy  to  deduce  the  amount  of  work  to 
be  transmitted  to  it.  But  this  force  cannot  be  exactly  deter- 


PUMPS.  119 

mined  ;  thus,  the  determination  of  friction  of  the  piston  against 
the  pump  barrel,  or  against  the  packing  that  it  passes  through 
(if  it  be  a  plunger),  is  necessarily  uncertain,  because  it  depends 
on  the  skill  of  the  constructor  ;  we  can  say  the  same  for  the 
losses  of  head  encountered  in  the  sucking  and  forcing  pipes,  on 
account  of  the  defect  of  permanence  and  uniformity  in  the 
motion.  However,  when  the  forcing  pipe  is  very  long,  we 
have  seen  that  it  is  well  so  to  arrange  matters  that  the  motion 
in  it  shall  be  uniform,  and  then  we  can  calculate  with  sufficient 
exactness  the  entire  head  £"  between  the  two  extremities  of 
this  pipe.  This  granted,  let  us  admit  first,  that  we  are  consid- 
ering a  double-stroke  pump  :  the  resultant  pressure  exerted  on 
the  piston  being  expressed  by  n  n  (H  -f  ?  +  ?')?  the  force  to 
be  transmitted  to  it  will  be  represented  by  n  ^  (H  -f  £")  -f  F. 
We  will  include  in  the  additive  term  F  the  friction  against  the 
barrel  of  the  pump  and  the  packing,  the  excess  of  %  +  £'  over 
£",  and  lastly  inertia.  The  total  work  of  this  force,  in  a  linear 
distance  Z,  divided  into  elements  d  x,  will  be 


Moreover,  n  I  represents  very  nearly  the  volume  of  water  raised 
by  one  stroke  of  the  piston  ;  if  then  we  wish  the  work  expend- 
ed in  raising  each  cubic  yard  of  water  to  the  height  H,  we  will 
have  to  calculate  the  quantity 


In  practice,  on  account  of  the  difficulty  of  determining  exactly 


the  integral^/   Fdx,  we  simply  multiply  the  term  n  (H 


-f 


by  a  co-efficient  n  such  as  1.10  or  1.15  or  1.20,  according  to 
the  greater  or  less  perfectness  of  the  machine. 

If  the  pump  were  one  of  single  stroke,  we  would  obtain  the 


120  PUMPS. 

same  expression  for  the  work,  by  adding  together  two  consecu- 
tive strokes  of  the  piston. 

When  we  wish  to  determine  the  amount  of  horse-power  to 
be  transmitted  to  the  piston,  we  must  also  know  the  mean  velo- 
city u  of  the  piston.  We  easily  deduce  the  mean  amount 
pumped  up  per  second  £2  u,  if  the  pump  is  one  of  double  stroke, 

or  — —  for  a  single-stroke  pump ;  we  multiply  this  amount  by 

n  n  (H  +  £") ;  dividing  finally  by  75,  we  have  the  horse-power 
sought. 

(c)  Meow,  velocity  of  the  piston  ;  delivery  of  pumps. — The 
mean  velocity  of  the  piston  ought  not  to  be  very  small,  because, 
in  order  to  pump  up  any  considerable  quantity  of  water,  we 
would  have  to  make  the  body  of  the  pump  and  the  lifting  pipe 
very  large,  which  would  increase  the  cost  of  construction.  But 
too  great  a  velocity  possesses  also  great  inconveniences :  firstly, 
we  increase  the  losses  of  head  in  a  very  rapid  proportion ;  then 
it  may  happen  that  the  water  furnished  by  the  sucking  pipe 
may  not  come  up  sufficiently  fast  to  follow  the  piston,  and  that 
the  pump  barrel  may  not  be  filled  at  each  stroke,  which  would 
cause  a  loss  in  the  delivery,  and  a  shock  on  the  return  of  the 
piston  in  an  opposite  direction.  On  account  of  the  difficulty 
of  determining  exactly  the  velocity  of  the  water  sucked  up,  we 
ordinarily  adopt  a  mean  velocity  of  the  piston,  about  Om.20  per 
second ;  we  rarely  go  so  high  as  Om.30.  It  is  clear  that  the 
limit  may  be  increased  in  proportion  as  the  piston  moves  to  a 
less  height  above  the  basin  from  which  the  water  is  drawn,  and 
the  more  care  that  has  been  taken  to  avoid  losses  of  head  in 
the  sucking  pipe. 

The  piston  having  a  stroke  the  length  of  which  is  I  and  the 
cross  section  n,  describes,  during  one  of  the  periods  employed 
in  raising  water  to  the  upper  basin,  a  volume  ft  I ;  this  volume 


PUMPS.  121 

would  also  be  that  of  the  water  raised  during  the  same  time, 
if  there  were  no  leakage  around  the  valves  between  the  piston 
and  the  hollow  cylinder  in  which  it  moves.  On  this  account, 
the  volume  raised  varies  from  0.75  n  I  to  Q  I ;  the  co-efficient 
by  which  the  volume  described  by  the  piston  is  to  be  affected 
varies  with  the  care  displayed  in  the  construction  and  in  keep- 
ing the  pump  in  good  order ;  under  ordinary  circumstances,  we 
may  take  it  from  0.90  to  0.92. 


SPIRAL    NORIA. 


23.  Spiral  Noria. — This  wheel  consists  essentially  of  a  hori- 
zontal arbor  O  (Fig.  20),  to  which  are  fastened  a  certain  num- 


FIG.  20. 


ber  of  cylindrical  surfaces  having  their  generatrices  parallel  to 
the  axis ;  the  right  sections  of  these  cylinders  are  involutes  of 
a  circle.  The  space  between  two  consecutive  cylinders  thus 
forms  a  canal  with  a  constant  breadth,  as  well  in  the  direction 
of  the  normal  as  perpendicularly  to  the  plane  of  the  figure. 
One  of  the  canals,  for  example,  will  have  its  outer  opening  at 
A  B  and  the  other  at  I G.  The  entire  system  turns  around 
the  axis  O,  in  the  direction  of  the  arrow-head ;  the  centre  O  is 
above  the  basin  whence  the  water  is  drawn,  and  the  level  of 
this  basin  covers  the  lower  part  of  the  wheel  more  or  less. 
During  the  time  that  the  opening  A  B  is  wholly  or  partially 


SPIRAL   NORIA.  123 

below  the  surface  level  of  the  basin,  a  certain  quantity  of  water 
enters  the  canal  A  B  I  Gr,  by  the  effect  of  the  rotation  ;  the  ro- 
tation continuing.  A  B  rises  until  finally  it  comes  directly  over 

1  G ;  then  the  water  taken  in  flows  into  I  G  through  holes  left 
open  all  around  the  arbor,  and  falls  into  a  canal  which  leads  it 
off  to  the  basin  that  is  to  receive  it. 

We  will  propose  two  questions:  1st,  a  given  spiral  noria 
turns  with  a  known  angular  velocity,  and  occupies  a  given  po- 
sition with  respect  to  the  lower  basin ;  what  will  be  the  amount 
raised  per  second  3  2d,  what  will  be  the  work  that  the  motor 
will  have  to  transmit  to  it  ? 

Let  us  call  S  the  section  A  B  projected  on  the  plane  passing 
through  the  axis  O  and  the  centre  of  A  B ;  N  the  number  of 
revolutions  of  the  wheel  per  minute ;  n  the  number  of  invo- 
lutes ;  H  the  height  O  C  of  the  point  O  above  the  level  of  the 
water  to  be  raised ;  r1 ',  r"  the  distances  of  the  points  A  and  B 
from  the  axis  of  rotation.  The  point  A  will  describe  under- 
neath the  water  an  arc  D  A  D',  of  which  we  will  designate  the 
angle  at  the  centre  by  2  a ;  in  like  manner  the  point  B  will  de- 
scribe the  arc  E  B  E',  corresponding  to  the  angle  at  the  centre 

2  /3.     First  we  shall  have 

H  H 


COS  a  =  — : , 

r'  r'" 

arc  D  A  D'  =  2  r'  «  =  2  X  cos  "%-, 

arc  E  B  E'  =  2  r"  ft  =  2  r"  cos  ~^; 

the  arc  described  below  the  water  by  the  centre  of  A  B  differ- 
ing little  from  the  mean  1  (D  A  D'  +  E  B  E')  will  then  be  ex- 

a 

pressed  by 

r'  cos     — r  +  r"  cos      -^  =  L. 
1*  r" 


124:  SPIRAL   NOKIA. 

Now,  the  volume  that  entered  at  the  opening  A  B  is  E  D  E'  D', 
or  the  product  of  this  arc  by  the  perpendicular  section  S ;  then, 
since  there  are  n  canals  that  raise  the  same  volume  in  each 
revolution  of  •  the  wheel,  the  volume  raised  will  be,  per  revolu- 
tion, n  L  S  ;  finally,  the  number  of  revolutions  per  second  being 

N 

— ,  the  amount  Q  raised  by  the  wheel  in  the  same  time  will  be 

t>0 

expressed  by  — -  N  n  L  S,  that  is,  we  ought  to  have 
60 

Q  =  lN«S(r'cos    ~^  +  r"  cos  ~^ 

But  this  calculation  supposes  that  there  enters,  during  each 
element  of  time  d  t  through  the  opening  A  B,  a  volume  of 
water  equal  to  that  generated  by  A  B  in  the  same  time ;  in 
this  way  the  contraction  which  the  liquid  may  experience  on 
entering  is  not  considered,  nor  is  the  motion  communicated  to 
the  surrounding  water,  which,  up  to  a  certain  point,  may  give 
way  before  the  surface  A  B,  instead  of  crossing  it.  For  these 
reasons  it  would  be  well  in  practice  to  admit  a  certain  reduc- 
tion in  the  value  of  Q  above  given  ;  we  could  effect  it,  for  ex- 
ample, by  a  co-efficient  which  we  will  value,  at  a  rough  esti- 
mate, at  0.80,  for  want  of  exact  experiments  on  this  subject. 

Here  is  an  example  for  calculating  Q.  Let  N"  =  12,  n  =  4r, 
r'  =  2m.50,  r"  =  3m.O,  H  =  2m.O,  S  =  0"*  m.17.  We  shall  have 

5  =  0.8000,    cos    "-?  =  0.410  1 ; 
r  r  2 

5=0.6667,    cos  "2  =  0.535^; 

S  cos  -^  =  r"  cos   "^5  =  ~  (1.025  +  1.605)  =  4.131 ; 
r  r         A 

whence  we  deduce 

Q  =  Omo.562, 


SPIRAL    NOKIA.  125 

a  number  which  would  be  reduced  to  Gmc.4-5  about,  by  multi- 
plying by  0.80. 

As  to  the  motive  work  to  be  expended  in  raising  a  certain 
weight  P  of  water,  it  is  composed :  1st,  of  the  work  P  H  des- 
tined to  oyercome  that  of  the  weight ;  2d,  of  the  work  of  fric- 
tion on  the  trunions  and  shoulders  of  the  arbor  O,  which  can 
be  determined  by  means  of  known  formulae ;  3d,  of  the  work 
necessary  to  overcome  the  friction  of  the  water  against  the 
solid  walls  with  which  it  is  in  contact ;  this  work  being  very 
slight,  if  the  involutes  form  tolerably  large  channels  ;  4th,  the 
work  necessary  to  give  to  the  water  the  absolute  velocity  with 
which  it  leaves  the  wheel.  This  last  work  will  also  be  very 
slight,  if  we  take  care  to  make  the  wheel  turn  slowly ;  for  the 
lowest  point  of  any  involute  whatever  being  always  on  the 
vertical  through  1,  we  see  that  the  water  that  has  already  en- 
tered the  interior  of  the  canal  A  B  I  G,  and  that  which  will 
still  enter  in  the  course  of  the  same  revolution,  will  only  be 
completely  emptied  out  after  an  entire  revolution,  reckoning 
from  the  position  indicated  by  the  figure.  The  water  rises 
then  with  little  absolute  velocity  into  the  machine,  and  conse- 
quently a  small  portion  of  the  motive  work  is  employed  in 
giving  to  it  an  unproductive  living  force.  But  it  must  not  be 
forgotten  that  this  supposes  slowness  of  revolution  around  the 
axis  O. 

To  sum  up,  we  will  calculate  the  first  two  portions  of  the 
motive  work,  which  are  the  most  important,  and  in  order  to 
account  approximately  for  the  other  two,  we  will  multiply  the 
sum  of  the  calculated  portions  by  a  co-efficient  a  little  greater 
than  unity. 

The  first  idea  of  the  noria  is  very  old,  since  Yitruvius  speaks 
of  a  similar  machine;  it  was  Lafaye  who,  in  1717,  proposed 
giving  it  the  form  we  have  described  above.  This  machine 


126  SPIRAL    NORIA. 

seems  susceptible  of  a  very  good  delivery,  and  is  well  adapted 
to  raising  large  volumes  of  water ;  but  the  height  to  which  the 
water  is  raised,  always  less  than  the  radius  of  the  wheel,  is 
necessarily  very  limited ;  besides,  this  wheel  is  heavy,  and  on 
this  account  hard  to  transport. 


CENTRIFUGAL    PUMP. 


24.  Lifting  turbines  ;  centrifugal  pump. — The  greater  part 
of  the  machines  which  are  used  to  turn  the  motive  power  of  a 
head  of  water  to  account,  can,  with  a  few  changes,  be  converted 
into  machines  for  raising  water,  and  the  reverse.  Thus,  for 
example,  if  a  breast-wheel,  set  in  a  water-course,  receives  a 
motion  about  its  horizontal  axis,  by  the  action  of  any  motor,  so 
that  the  floats  may  ascend  the  circular  flume,  these  floats  will 
carry  up  with  them  the  water  from  the  tail  race  and  throw  it 
into  the  head  race :  we  would  then  obtain,  in  principle,  the 
lifting  wheel.  In  like  manner,  let  us  take  one  of  Fourneyron's 
turbines,  and  make  the  intervals  between  the  directing  parti- 
tions communicate  directly  with  the  tail  race,  and  let  the  outer 
orifices  of  the  turbine  open  into  a  compartment  from  which  the 
ascent  pipe  leads ;  when  a  motion  of  rotation  is  impressed  on 
the  wheel,  the  water  contained  in  the  floats  will  be  urged  to- 
ward the  exterior  by  the  centrifugal  force,  and  will  reach  the 
enclosed  compartment  with  an  excess  of  pressure  which  will 
cause  it  to  ascend  the  pipe  to  a  certain  height,  the  greater  as 
the  rotation  becomes  more  rapid.  If  the  pipe  is  not  too  high, 
a  delivery  of  water  will  take  place  at  its  end ;  and  this,  more- 
over, will  be  continuous,  the  water  thrown  out  by  the  centrifu- 
gal force  being  incessantly  replaced  by  that  from  the  tail  race, 
which  tends  to  fill  up  the  empty  space  between  the  partitions. 

The  theory  of  such  a  turbine,  which  we  might  call  a  lifting 
turbine,  resembles  very  closely  that  of  (No.  15).  But  as  the 


128 


CENTRIFUGAL    PUMP. 


machine  there  discussed  has  not  as  yet  been  set  up  or  experi- 
mented upon,  it  need  no  longer  be  dwelt  upon.  We  will  pass 
to  the  study  of  a  pump  called  the  centrifugal  pump,  which 
belongs  to  the  same  class  of  machines,  but  which  bears,  how- 
ever, a  greater  resemblance  to  reaction  wheels. 

A  wheel  composed  of  a  series  of  cylindrical  floats,  such  as 
B  C  (Fig.  21),  assembled  between  two  annular  plates,  is  caused 


FIG.  81. 


to  turn  around  a  horizontal  axis  projected  at  A.  The  water, 
from  the  basin  to  be  emptied,  comes  freely  within  the  circle 
A  B,  which  limits  the  floats  on  the  inside,  either  because  the 
centre  A  is  a  little  below  the  level  N  N  of  this  bay,  or  by 
means  of  suction  pipes.  The  motion  of  rotation  impressed  on 
this  wheel  drives  the  water  from  the  canals  B  C,  B'  C',  into  the 
annular  space  D,  where  it  acquires  a  pressure  sufficient  to  drive  it 
up  the  pipe  E,  the  only  means  of  escape  open  to  it,  and  by  which 
it  reaches  the  upper  basin.  The  angular  velocity  of  the  arbor 
A  being  known,  as  well  as  all  the  dimensions  of  the  machine, 
and  its  position  relatively  to  the  basins  of  departure  and  arrival, 


CENTRIFUGAL   PUMP.  129 

we  can  find  the  amount  of  water  pumped  up  per  second,  the 
motive  work  that  it  requires,  and  its  delivery. 

To  show  this,  let  us  call 

H  the  difference  of  level  between  the  two  basins ; 

h  the  depth  of  the  centre  A  below  the  level  of  the  lower ; 

r  the  exterior  radius  A  C  of  the  wheel ; 

ft  the  distance  apart  of  the  two  annular  plates,  which  confine 
the  floats ; 

w  the  angular  velocity  of  the  arbor  A ; 

v  the  absolute  velocity  of  the  water  when  it  leaves  the  floats ; 

u  the  velocity  w  r  at  the  outer  circumference  of  the  wheel ; 

w  the  relative  velocity  of  the  water  at  the  same  point ; 

7  the  acute  angle  formed  by  the  velocities  w  and  u — that  is, 
the  angle  at  which  the  floats  cut  the  outer  circumference ; 

p  the  pressure  of  the  water  at  its  point  of  entrance  into  the 
interval  between  the  floats  ; 

p'  its  pressure  at  the  point  of  exit ; 

pa  the  atmospheric  pressure ; 

n  the  weight  of  the  cubic  metre  of  water. 

We  will  begin  by  simplifying  the  question  a  little  by  means 
of  a  few  hypotheses.  First,  we  will  neglect  the  absolute  velo- 
city of  the  water  in  the  ascent  pipe  and  in  the  conduit  which 
conveys  it  to  the  floats,  which  may  be  allowed  if  the  cross  sec- 
tions of  these  conduits  are  sufticiently  large  relatively  to  the 
volume  pumped  out.  The  radius  A  B,  however,  should  still 
be  sufficiently  small  so  that  the  velocity  of  rotation  of  the  point 
B  may  be  neglected ;  in  other  words,  we  will  consider  the  in- 
troduction of  the  water  into  the  wheel  as  taking  place  along 
the  axis,  without  any  velocity  occasioned  by  the  motion,  and 
consequently  without  any  relative  velocity.  Secondly,  we  will 
conduct  our  argument  as  though  the  water  were  displaced  hori- 
zontally in  its  passage  across  the  wheel ;  the  height  of  this  last 


130  CENTRIFUGAL   PUMP. 

will  be  supposed  slight  relatively  to  H,  so  that  it  can  be  left 
out  of  account.  Besides,  nothing  in  practice  would  prevent 
our  assuming  the  arbor  A  as  vertical  ;  but  this  would  be  a 
matter  of  very  little  importance  in  the  result. 

This  granted,  the  pressure  varying  according  to  the  hydro- 
static law  from  the  lower  basin  to  the  point  of  entrance,  and 
from  the  point  of  exit  to  the  other  basin,  we  will  have 


whence,  by  subtraction, 


n 

Now,  if  we  apply  Bernoulli's  theorem  to  the  relative  motion 
of  a  molecule  'following  the  curve  B  C,  the  fictitious  gain  of 

head  will  be  expressed  by  —  —  or  —  —  ,  and  we  shall  find 

2  g       2  g 

w*     _  p  —p'       u* 
~2~7=     ~~rT   *~2~7 

or  else 


<  ...  (1) 
an  equation  giving  w  since  u  is  known.  This  first  result  gives 
the  means  of  calculating  the  amount  Q  pumped  up  in  each 
second.  In  fact,  the  water  leaving  the  floats  cuts  a  cylindrical 
surface  2  it  ~b  r  at  an  angle  7  and  with  the  relative  velocity  w  • 
then  the  total  orifice  of  exit,  measured  perpendicularly  to  w,  is 
2  if  1)  r  sin  7,  and  consequently 

Q  =  2  if  b  r  w  sin  7.    .     .     .     (2) 

The  motive  work  consumed  per  second  in  making  the  wheel 
turn  includes  first  the  work  n  Q  H  ;  then  the  water  reaching 
the  annular  space  D  with  a  velocity  0,  this  is  lost  in  useless 

v9 
disturbance  ;  whence  there  results  a  molecular  work  n  Q  —  —  . 


CENTRIFUGAL   PUMP.  131 

Thus  then,  throwing  out  of  account  the  other  frictions,  the 

work  expended  per  second  will  be  n  Q  (H  4-  —-\  ;   and  as  the 

2t  o '/ 

useful  work  is  only  n  Q  H,  the  effective  delivery  fx  will  have  for 


its  value 


21  xx  TT 

g  2^H 

There  remains  to  determine  v  /  now  v  is  the  resultant  of  w  and 
u,  hence  we  have 

v9  =  u3  +  w9  +  2  u  w  cos  7, 
or  from  eq.  (1) 


v*  =  —  2  g H  +  2  u*  —  2  u  cos  7  ^  —  2  ^H  +  w5 (4) 

Equations  (1),  (2),  (3),  and  (4)  give  the  means  of  solving  with- 
out difficulty  the  questions  proposed. 

Let  us  again  see  by  what  means  we  could  obtain  the  greatest 
possible  result  of  the  motive  power.  Expression  (3)  for  the 
effective  delivery  becomes,  substituting  for  v  its  value,  and 

,  .  u 

making  —   ==-  =  «?. 


x  —  x  cos  7 

we  shall  then  have  the  maximum  of  f*,  considered  as  a  function 
of  a?,  in  seeking  the  minimum  of  the  denominator,  or,  what 

amounts  to  the  same,  the  minimum  of  -.      We  shall   conduct 
this  research  as  in  (No.  21) ;  we  will  write 

a?2  —  -  —  x  cos  7   V  x*  —  2 
or,  by  making  the  radical  disappear  and  transposing, 

x*  sin3  7  —  2  a?a  ( cos9  7)  -\ — 3  =  0. 

9 


132  CENTRIFUGAL    PUMP. 

Now  fA  can  only  receive  values  that,  substituted  in  this  biquad- 
ratic equation,  will  give  a?a  real  and  positive;  hence  we  have 

(  --  cos3  7^  --  -  sin8  7  >  0, 
\  f*  /         M- 

or  successively 

1  2 

—  cos8  7  --  cos8  7  +  cos4  7  >  0, 


As  sin  7  and  --  1  are  positive  quantities,  we  can  extract  the 
M* 

square  root  of  both  members  and  place 

—  1  >  sin  7,  or  -  >  1  -f  sin  7  ; 

fX  fX 

the  minimum  of  -  has  then  for  its  value  1  +  sin  7,  and  the 

limit  of  the  effective  delivery  ^.  will  be  -  --  :  —  .      The  corre- 

1  4-  sin  7 

spending  value  xt  of  x  is  obtained  from  the  above  biquadratic 
equation,  which  gives 

-  —  cos8  7 
a  _/Xj  _  1  +  sm  7  —  cos  7  __  1  +  sin  7 

sin2  7  sin2  7  sin  7 

Thus  the  most  favorable  velocity  u  for  the  effective  delivery  is 
obtained  from  this  equation 


sin  7 
whence  we  get  the  angular  velocity  w  =  -,  and  the  number  of 

revolutions  per  minute  N  =  -  .    The  effective  delivery  being 


CENTKIFUGAL    PUMP.  133 

then  -  —  —  r  —  ,  we  may  be  tempted,  in  order  to  increase  it,  to 

JL   ""f**  Sill  j 

make  7  very  small  ;  but  we  see  that  the  velocities  u  and  w 
would  become  very  great,  and  we  should  thus  lose  a  great  deal 
in  the  friction  of  the  water  against  the  floats.  Besides,  we 
have,  from  equations  (1)  and  (2), 

Q  =  2tf&7'sm  7  V  u*  —  2~<?H  =  2  *  I  r  V  g  H  sin  7.  V  X*  —  2~j 
the  amount  Q',  which  corresponds  to  the  maximum  effective 
delivery,  will  then  be 


Q'  =  2  *  IT  V~R.  sin 


sn  7 

=  2*br  V  gIL  V  sin  7  (1  —  sin  7). 

This  amount  reduces  to  zero  at  the  same  time  as  7  ;  when 
7  alone  varies,  Q'  becomes  a  maximum  for  sin  7  =  1  —  sin  7,  or 

sin  7  =  -  or  7   =  30°  ;    the  effective   delivery  is  then  --  - 

1+2 

2 

that  is  ^.     The  value  7  =  o  is  consequently  inadmissible  as  re- 
o 

ducing  the  amount  to  zero  ;  but  from  this  point  of  view  it  is 
not  well  to  go  beyond  7  —  30°.  On  the  other  hand,  this  last 
value  does  not  give  a  very  high  theoretical  effective  delivery; 
perhaps  the  best  thing  to  do  in  practice  would  be  to  take  7  be- 
tween 15  and  20  degrees.  For  7  =  15°,  for  example,  the  eifec- 

tive  delivery  increases  to  -  -  TT^TTH  —  0-794,  and  the  product 

1  +  0.2588 

t/'sTn  7  (1  —  sin  7)  is  decreased  to  0.438,  whereas  it  is  0.50  for 
7  =  30°  ;  it  is  a  diminution  that  could  be  compensated  for  by 
a  slight  increase  of  r  or  5. 

Instead  of  arranging  the  wheel  as  represented  in  Fig.  21,  we 
might  adopt  two  separate  canals,  as  in  Fig.  19.  In  this  case 
the  expression  for  the  amount  would  change,  and  the  angle  7 


134  CENTRIFUGAL   PUMP. 

might  become  zero ;  but,  whereas  u  cannot  increase  to  infinity, 

the  value  — 2L—-J1  would  no  longer  be  admissible  for  — =.  and 
sin  7  g  IT 

we  should  have  to  depart  more  or  less  from  the  limit  of  the 
effective  delivery.  Furthermore,  for  an  equal  expenditure,  we 
should  probably  lose  more  in  friction. 


AUTHORITIES  ON  WATER  WHEELS. 

Experiences  sur  les  Roues  Hydrauliques  a  aubes  planes,  et  sur  les  Roues 
Hydrauliques  a  augets,  by  Morin. 

Experiments  made  by  the  Committee  of  the  Franklin  Institute  on  Water 
Wheels,  in  1829-30.  See  Journal  of  the  Franklin  Institute,  3d  Series,  Vol. 
I,  pp.  149,  154,  &c.,  and  Vol.  II.,  p.  2. 

Experiments  on  Water  Wheels,  by  Elwood  Morris.  See  Jour.  Frank. 
Ins.,  3d  Series,  Vol.  IV.,  p.  222. 

Memoire  sur  les  Roues  Hydrauliques  a  aubes  courses,  mues  par  dessous,  by 
Poncelet. 

Experiences  sur  les  Roues  Hydrauliques  a  axe  vertical,  appelees  Turbines,  by 
Morin. 

Experiments  on  the  Turbines  of  Fourneyron.  by  Elwood  Morris.  See 
Jour.  Frank.  Ins.,  Dec.  1843,  and  Jour.  Frank.  Ins.,  3d  Series,  Vol.  IV.,  p. 
303. 

Lowell  Hydraulic  Experiments,  2d  Ed.,  1868,  by  Jas.  B.  Francis. 


APPENDIX. 


Comparative  Table  of  French  and  United  States  Measures. 

Pounds  avoirdupois  in  a  kilogramme 2.2 

Inch  in  a  millimetre 0.039 

Inch  in  a  centimetre 0.393 

Inch  in  a  decimetre 3.937 

Feet  in  a  metre 3.280 

Yard  in  a  metre < 1.093 

Square  feet  in  a  square  metre 10.7643 

Cubic  inch  in  a  cubic  centimetre 0.061 

Cubic  feet  in  a  cubic  metre 35.316 

Quart  in  a  litre 1.0567 

ISToTE. — A  cubic  metre  of  distilled  water  weighs  one  thousand 
kilogrammes. 

The  litre  contains  one  cubic  decimetre  of  distilled  water,  and 
weighs  one  kilogramme. 

The  horse-power  of  the  French  is  75  kilogram  metres,  equiva- 
lent to  542|-  foot-pounds  per  second  ;  the  English  horse-power 
being  550  foot-pounds  per  second.  The  value  of  g  is  9.81 
metres;  and  that  of  the  height  of  a  column  of  water  equivalent 
to  the  atmospheric  pressure  is  taken  at  10.33  metres. 


Note  A.     Art.  1. 

The  formula  given  in  this  article  is  to  be  found,  with  the 
exception  of  variations  in  the  notation,  in  all  works  of  applied 


136  APPENDIX. 

mechanics  in  which  the  subject  of  the  theory  of  machines  is 
discussed  (see,  for  example,  Moseley's  Engineering  and  Archi- 
tecture, Am.  Ed.,  p.  146).  The  only  term  in  it  which  is  not 
generally  found  in  other  works  is  the  one  (H  —  H0)  2  my,  which 
expresses  the  work  expended  in  overcoming  the  weight  of  any 
part  of  the  machine,  when  its  centre  of  gravity  is  raised  from 
one  level  to  another,  represented  by  the  vertical  height  (H  — 
H0),  and  the  corresponding  work  by  (H  —  H0)  2  m  g  /  as,  for 
example,  in  the  case  of  a  wheel  revolving  on  a  horizontal  axle, 
the  axis  of  which  does  not  coincide  with  its  centre  of  gravity ; 
or  in  that  of  a  revolving  crank ;  in  both  of  which  cases  the 
work  expended  will  be  equal  to  the  product  of  the  weight 
2  m  g  raised,  and  the  vertical  height  (H  —  H0)  passed  over  by 
its  centre  of  gravity.  But,  in  all  like  cases,  as  in  the  descent  of 
the  centre  of  gravity  from  its  highest  to  its  lowest  position,  the 
same  amount  of  work  will  be  restored  by  the  action  of  gravity, 
the  total  work  expended  will  be  zero  for  each  revolution,  and 
the  term  (H  —  H0)  2  m  g  will  disappear  from  the  formula. 


Note  B.     Art.  2. 

1  U2  —  Ua 

The  equation  H (te+  tj) =  o  is  the  modified 

9  2g 

form  of  what  is  known  as  Bernoulli's  theorem  as  applied  to 
the  case  treated  of  in  Art.  2. 

This  theorem,  applied  to  the  phenomena  of  the  flow  of  a 
heavy  homogeneous  fluid,  may  be  generally  thus  stated:  The 
increase  of  height  due  to  the  velocity  is  equal  to  the  difference 
between  the  effective  head  and  the  loss  of  head. 

In  this  case  H  is  the  effective  head  ; (te  +  tf/)  is  the  loss 

of  head  from  the  dynamical  effect  te  imparted  to  the  wheel  by 


APPENDIX.  137 

the  action  of  the  water,  and  the  work  tf  due  to  the  various  sec- 
ondary resistances ;  and  the  term ~L 1  the  height  due  to 

the  velocity. 

Multiplying  each  member  of  the  above  equation  by  g,  we 
obtain 

9         ~  *e  ~  tf  =  g  ^       ~  £  U  <» 

or,  in  other  words,  the  modified  expression  of  the  general  for- 
mula Note  A,  as  applied  to  this  case. 

See  Bresse.     Mecanique  Appliquee.      Vol.  2,  Nos.  12  and 
->.  23  and  30. 


Note  C.    Art.  4. 
In  the  equation 


which  expresses  the  force  applied  horizontally  at  the  centre  of 
the  submerged  portion  of  the  bucket,  the  second  term  of  the 

second  member  -  n  J  (A/a  —  A2),  represents  the  diminution  of 

the  force  imparted  to  the  wheel  by  the  current,  arising  from 
the  increase  of  depth  of  the  water  as  it  leaves  the  wheel,  or  by 
the  back  water  ;  or,  in  other  words,  the  difference  of  level  be- 
tween the  point  C,  before  the  depth  of  the  current  is  affected 
by  the  action  of  the  wheel,  and  the  point  E,  where  the  depth 
of  the  current  has  increased  from  the  back  water.  This  differ- 
ence of  level  receives  the  name  of  a  surface  fall. 

The  relations  existing  between  the  two  terms  of  the  second 
member  of  the  equation,  leaving  out  of  consideration  the  action 
of  the  wheel,  may  be  established  in  the  following  manner. 

Considering  the  portion  of  the  current  comprised  between  the 


138  APPENDIX. 

two  sections  C  B,  E  F  (Fig.  3),  at  a  short  distance  apart,  be- 
tween which  the  surface  fall  takes  place,  we  can  apply  to  the 
liquid  system  C  B  E  F,  comprised  between  these  sections,  the 
theorem  of  the  quantities  of  motion  projected  on  the  axis  of 
the  current^  which,  in  the  present  case,  may  be  regarded  as 
horizontal.  Now,  during  a  very  short  interval  of  time  0,  the 
system  C  B  E  F  will  have  changed  its  position  to  C7  B7  E7  F7, 
and,  in  virtue  of  the  supposed  permanency  of  the  motion,  each 


FIG.  8. 


point  of  the  intermediate  portion  C7B7EF  will  have  equal 
masses  moving  with  the  same  velocity  at  the  beginning  and 
ending  of  the  time  6 ;  the  variation  in  the  projected  quantity  of 
motion  of  the  system  C  B  E  F,  during  the  time  0,  will  therefore 
be  equal  to  the  quantity  of  motion  of  the  portion  included  be- 
tween the  final  sections  E  F,  E7  F7,  and  that  comprised  between 
the  initial  sections  C  B,  C7  B7. 

To  find  these  quantities  of  motion.  Represent  by  w7  a  super- 
ficial element  of  the  section  E  F,  and  by  vr  the  velocity  of  the 
fluid  thread  which  flows  through  it ;  v'  6  will  then  be  the  length 
of  this  thread  for  the  time  0,  between  the  sections  E  F  and 
E7  F7 ;  and  w'  v'  6  will  be  the  volume  of  the  thread  which  has 
w7  for  its  base  and  v'  6  for  its"  length.  Representing  by  n  the 

weight  of  a  cubic  metre  of  the  liquid,  —  w7  v'  6  will  be  the  cor- 

£/ 

responding  mass  of  this  volume,  and  —  w7  v'*  6  its  quantity  of 

it 

motion  ;  and,  designating  by  s  the  sum  of  all  the  elements  w', 


APPENDIX.  139 

—  SwVad  will   be   the   quantity   of    motion   of  the  portion 

E  F  E'  F'  of  the  liquid  comprised  between  the  two  final  sec- 
tions. In  like  manner,  v  being  the  velocity  with  which  each 
thread  flows  through  an  element  w  of  the  section  B  C,  the  quan- 
tity of  motion  of  the  portion  of  liquid  between  the  sections  B  C 

and  B'  C'  will  be  expressed  by  —  s  w  tf  &.     The  increase,  there- 

y 

fore,  in  the  quantity  of  motion  during  the  time  6  will  be  ex- 
pressed by 

5L  (s  «'  v'*  &  —  S  w  V9  6} 

V 

But  as  v  and  vf  may  be  assumed  as  sensibly  equal  to  the  mean 
velocities  of  the  current  in  sections  E  F,  CB,  then  Sw'0'  and 
s  w  v  will  be  the  volumes  corresponding  to  these  velocities ;  and 

-  s  w'  v'  and  —  s  w  v  the  corresponding  masses.     But  since, 

y  & 

from  the  permanency  of  the  motion  w'  v'  =  u  v,  the  expression 

for  the  increase  of  the  quantity  of  motion,  for  the  time  6  will 
therefore  take  the  form 

Pa,  ,         x 

g      («'-"), 

in  which  P  represents  the  weight  of  the  water  expended  in  each 

p 

second,  and  —  its  corresponding  mass. 

y 

The  expression  here  found  is  equal  to  the  sum  of  the  impul- 
sions of  the  forces  exterior  to  the  liquid  system  considered 
during  the  time  6,  also  projected  on  the  horizontal  axis  of  the 
current.  From  the  form  given  to  the  section  of  the  race,  which 
is  rectangular,  the  direction  of  the  axis  of  the  current,  which  is 
assumed  as  horizontal,  between  the  extreme  sections,  and  the 
short  distance  between  these  sections,  the  only  impulsions  of  the 
pressures  upon  the  liquid  system  are  those  on  the  sections  E  F 


140  APPENDIX. 

and  C  B.  Representing,  then,  by  b  the  breadth  of  the  sections, 
by  h'  and  h  their  respective  depths,  their  respective  areas  will 
be  expressed  by  5  h'  and  b  h  ;  the  pressures  on  these  areas  will 

be  n  ~b  h'  x  -  h'  and  n  5  h  x  -  h  ;  and  for  the  respective  pro- 
jected sums  of  the  impulsions  of  these  pressures,  during  the 

time  6,  we  shall  have  -  n  b  &  hf*  and  -  n  b  6  A9,  since  from  the 

2  2 

circumstances  of  the  motion  the  pressures  follow  the  hydro- 
static law.  The  impulsion  in  the  direction  of  the  motion  will 

therefore  be  expressed  by  -  n  o  &  (A3  —  A/a),  from  which  we  ob- 
tain 

—  6  (vr  -  v)  =  -Tib  6(h*  -  A"). 
9  2 

to  express  the  relation  in  question. 

See  Bresse.     Mecanique  Appliquee.     Vol.  2,  No.  83,  p.  245. 


Note  D.     Art.  9. 
The  term  Cfl  —  ^j^j  —  0,  which  expresses  the   arc  inter- 

\       2  _L\y 


cepted  between  two  buckets,  taken  at  the  middle  point  of  their 
depth,  is  obtained  as  follows  : 

R  being  the  exterior  radius  of  the  wheel,  corresponding  to 
the  arc  C,  the  radius  of  the  arc  at  the  middle  point  of  the 

bucket  will  be  R  --  p  ;   calling  x  the  arc  corresponding  to 

2 

this  radius,  we  have 


and  for  the  arc  intercepted  between  the  two  buckets  at  their 
middle  point  the  expression  above. 


APPENDIX.  141 

Note  E. 

By  the  courteous  permission  of  JAMES  B.  FRANCIS,  Esq.,  granted  through 
G-en.  JOHN  C.  PALFREY,  the  following  extracts  were  taken  from  the  val- 
uable work  of  Mr.  Francis,  under  the  title  of  "  LOWELL  HYDRAULIC  EXPERI- 
MENTS." 

A  VAST  amount  of  ingenuity  has  been  expended  by  intel- 
ligent millwrights  on  turbines ;  and  it  was  said,  several  years 
since,  that  not  less  than  three  hundred  patents  relating  to  them 
had  been  granted  by  the  United  States  Government.  They 
continue,  perhaps,  as  much  as  ever  to  be  the  subject  of  almost 
innumerable  modifications.  Within  a  few  years  there  has  been 
a  manifest  improvement  in  them,  and  there  are  now  several 
varieties  in  use,  in  which  the  wheels  themselves  are  of  simple 
forms,  and  of  single  pieces  of  cast  iron,  giving  a  useful  effect 
approaching  sixty  per  cent,  of  the  power  expended. 

In  the  journal  of  the  Franklin  Institute,  Mr.  Morris  also 
published  an  account  of  a  series  of  experiments,  by  himself,  on 
two  turbines  constructed  from  his  own  designs,  and  then  ope- 
rating in  the  neighborhood  of  Philadelphia. 

The  experiments  on  one  of  these  wheels  indicate  a  useful 
effect  of  seventy-five  per  cent,  of  the  power  expended,  a  result 
as  good  as  that  claimed  for  the  practical  effect  of  the  best  over- 
shot wheels,  which  had  heretofore  in  this  country  been  con- 
sidered unapproachable  in  their  economical  use  of  water. 

In  the  year  1844,  Uriah  A.  Boyden,  Esq.,  an  eminent 
hydraulic  engineer  of  Massachusetts,  designed  a  turbine  of 
about  seventy-five  horse  power,  for  the  picking-house  of  the 
Appleton  Company's  cotton-mills,  at  Lowell,  in  Massachusetts, 
in  which  wheel  Mr.  Boyden  introduced  several  improvements 
of  great  value. 

The   performance  of  the  Appleton  Company's  turbine  was 


142  APPENDIX. 

carefully  ascertained  by  Mr.  Boyden,  and  its  effective  power, 
exclusive  of  that  required  to  carry  the  wheel  itself,  a  pair  of 
bevel  gears,  and  the  horizontal  shaft  carrying  the  friction-pulley 
of  a  Prony  dynamometer,  was  found  to  be  seventy-eight  per 
cent,  of  the  power  expended. 

In  the  year  1846,  Mr.  Boyden  superintended  the  construc- 
tion of  three  turbines,  of  about  one  hundred  and  ninety  horse- 
power each,  for  the  same  company.  By  the  terms  of  the  con- 
tract, Mr.  Boy  den's  compensation  depended  on  the  performance 
of  the  turbines ;  and  it  was  stipulated  that  two  of  them  should 
be  tested.  In  accordance  with  the  contract,  two  of  the  turbines 
were  tested,  a  very  perfect  apparatus  being  designed  by  Mr. 
Boyden  for  the  purpose,  consisting  essentially  of  a  Prony  dyna- 
mometer to  measure  the  useful  effects,  and  a  weir  to  gauge  the 
quantity  of  water  expended. 

The  observations  were  put  into  the  hands  of  the  author  for 
computation,  who  found  that  the  mean  maximum  effective 
power  for  the  two  turbines  tested  was  eighty-eight  per  cent,  of 
the  power  of  the  water  expended. 

According  to  the  terms  of  the  contract,  this  made  the  com- 
pensation for  engineering  services,  and  patent  rights  for  these 
three  wheels,  amount  to  fifty-two  hundred  dollars,  which  sum 
was  paid  by  the  Appleton  Company  without  objection. 

These  turbines  have  now  been  in  operation  about  eight 
years,  and  their  performance  has  been,  in  every  respect,  entirely 
satisfactory.  The  iron  work  for  these  wheels  was  constructed 
by  Messrs.  Gay  &  Silver,  at  their  machine-shop  at  North 
Chelmsford,  near  Lowell ;  the  workmanship  was  of  the  finest 
description,  and  of  a  delicacy  and  accuracy  altogether  unpre- 
cedented in  constructions  of  this  class. 

These  wheels,  of  course,  contained  Mr.  Boyden's  latest  im- 
provements, and  it  was  evidently  for  his  pecuniary  interest  that 


APPENDIX.  143 

the  wheels  should  be  as  perfect  as  possible,  without  much  regard 
to  cost.  The  principal  points  in  which  one  of  them  differs 
from  the  constructions  of  Fourneyron  are  as  follows : 

The  wooden  flume  conducting  the  water  immediately  to  the 
turbine  is  in  the  form  of  an  inverted  truncated  cone,  the  water 
being  introduced  into  the  upper  part  of  the  cone,  on  one  side  of  the 
axis  of  the  cone  (which  coincides  with  the  axis  of  the  turbine),  in 
such  a  manner  that  the  water,  as  it  descends  in  the  cone,  has  a 
gradually  increasing  velocity  and  a  spiral  motion  /  the  horizontal 
component  of  the  spiral  motion  being  in  the  direction  of  the 
motion  of  the  wheel.  This  horizontal  motion  is  derived  from 
the  necessary  velocity  with  which  the  water  enters  the  trun- 
cated cone;  and  the  arrangement  is  such  that,  if  perfectly  pro- 
portioned, there  would  be  no  loss  of  power  between  the  nearly 
still  water  in  the  principal  penstock  and  the  guides  or  leading 
curves  near  the  wheel,  except  from  the  friction  of  the  water 
against  the  walls  of  the  passages.  It  is  not  to  be  supposed  that 
the  construction  is  so  perfect  as  to  avoid  all  loss,  except  from 
friction ;  but  there  is,  without  doubt,  a  distinct  advantage  in 
this  arrangement  over  that  which  had  been  usually  adopted, 
and  where  no  attempt  had  been  made  to  avoid  sudden  changes 
of  direction  and  velocity. 

The  guides,  or  leading  curves  (Figs.  A,  B),  are  not  perpen- 
dicular, hut  a  little  inclined  backwards  from  the  motion  of  the 
wheel,  so  that  the  water,  descending  with  a  spiral  motion,  meets 
only  the  edges  of  the  guides.  This  leaning  of  the  guides  has 
also  another  valuable  effect :  when  the  regulating  gate  is  raised 
only  a  small  part  of  the  height  of  the  wheel,  the  guides  do  not 
completely  fulfil  their  office  of  directing  the  water,  the  water 
entering  the  wheel  more  nearly  in  the  direction  of  the  radius 
than  when  the  gate  is  fully  raised  ;  by  leaning  the  guides  it  will 
be  seen  the  ends  of  the  guides  near  the  wheel  are  inclined,  the 


APPENDIX. 


bottom  part  standing  farther  forward,  and  operating  more 
efficiently  in  directing  the  water  when  the  gate  is  partially 
raised,  than  if  the  guides  were  perpendicular. 

In  Fourneyron's  constructions  a  garniture  is  attached  to 
the  regulating  gate,  and  moves  with  it,  for  the  purpose  of  di- 
minishing the  contraction.  This,  considered  apart  from  the 
mechanical  difficulties,  is  probably  the  best  arrangement;  to 
be  perfect,  however,  theoretically,  this  garniture  should  be  of 
different  forms  for  different  heights  of  gate  ;  but  this  is  evi- 
dently impracticable. 

In  the  Appleton  turbine  the  garniture  is  attached  to  the 
guides,  the  gate  (at  least  the  lower  part  of  it)  being  a  simple  thin 
cylinder.  By  this  arrangement  the  gate  meets  with  much 
less  obstruction  to  its  motion  than  in  the  old  arrangement,  un- 
less the  parts  are  so  loosely  fitted  as  to  be  objectionable  ;  and  it 
is  believed  that  the  coefficient  of  effect,  for  a  partial  gate,  is 
proportionally  as  good  as  under  the  old  arrangement. 

On  the  outside  of  the  wheel  is  fitted  an  apparatus,  named  by 
Mr.  Boyden  the  Diffuser.  The  object  of  this  extremely  inter- 
esting invention  is  to  render  useful  a  part  of  the  power  other- 
wise entirely  lost,  in  consequence  of  the  water  leaving  the  wheel 
with  a  considerable  velocity.  It  consists,  essentially,  of  two 
stationary  rings  or  discs,  placed  concentrically  with  the  wheel, 
having  an  interior  diameter  a  very  little  larger  than  the  exte- 
rior diameter  of  the  wheel  ;  and  an  exterior  diameter  equal  to 
about  twice  that  of  the  wheel  ;  the  height  between  the  discs  at 
their  interior  circumference  is  a  very  little  greater  than  that  of 
the  orifices  in  the  exterior  circumference  of  the  wheel,  and  at 
the  exterior  circumference  of  the  discs  the  height  between  them 
is  about  twice  as  great  as  at  the  interior  circumference  ;  the  form 
of  the  surfaces  connecting  the  interior  and  exterior  circumfer- 
ences of  the  discs  is  gently  rounded,  the  first  elements  of  the 


APPENDIX.  145 

curves  near  the  interior  circumferences  being  nearly  horizon- 
tal. There  is  consequently  included  between  the  two  surfaces 
an  aperture  gradually  enlarging  from  the  exterior  circumference 
of  the  wheel  to  the  exterior  circumference  of  the  diffuser.  When 
the  regulating  gate  is  raised  to  its  full  height,  the  section  through 
which  the  water  passes  will  be  increased,  by  insensible  degrees, 
in  the  proportion  of  one  to  four,  and  if  the  velocity  is  uniform 
in  all  parts  of  the  diffuser  at  the  same  distance  from  the  wheel, 
the  velocity  of  the  water  will  be  diminished  in  the  same  pro- 
portion ;  or  its  velocity  on  leaving  the  diffuser  will  be  one-fourth 
of  that  at  its  entrance.  By  the  doctrine  of  living  forces,  the 
power  of  the  water  in  passing  through  the  diffuser  must,  there- 
fore, be  diminished  to  one-sixteenth  of  the  power  at  its  entrance. 
It  is  essential  to  the  proper  action  of  the  diffuser  that  it  should 
be  entirely  under  water,  and  the  power  rendered  useful  by  it 
is  expended  in  diminishing  the  pressure  against  the  water  issu- 
ing from  the  exterior  orifices  of  the  wheel ;  and  the  effect  pro- 
duced is  the  same  as  if  the  available  form  under  which  the 
turbine  is  acting  is  increased  a  certain  amount.  It  appears 
probable  that  a  diffuser  of  different  proportions  from  those  above 
indicated  would  operate  with  some  advantage  without  being 
submerged.  It  is  nearly  always  inconvenient  to  place  the 
wheel  entirely  below  low-water  mark ;  up  to  this  time,  however, 
all  that  have  been  fitted  up  with  a  diffuser  have  been  so  placed  ; 
and  indeed,  to  obtain  the  full  effect  of  a  fall  of  water,  it  appears 
essential,  even  when  a  diffuser  is  not  used,  that  the  wheel  should 
be  placed  below  the  lowest  level  to  which  the  water  falls  in  the 
wheel-pit,  when  the  wheel  is  in  operation. 

The  action  of  the  diffuser  depends  upon  similar  principles 
to  that  of  diverging  conical  tubes,  which,  when  of  certain  pro- 
portions, it  is  well  known,  increase  the  discharge ;  the  author 
has  not  met  with  any  experiments  on  tubes  of  this  form  dis- 


146  APPENDIX. 

charging  under  water  although  there  is  good  reason  to  believe 
that  tubes  of  greater  length  and  divergency  would  operate  more 
effectively  under  water  than  when  discharging  freely  in  the 
air,  and  that  results  might  be  obtained  that  are  now  deemed 
impossible  by  most  engineers. 

Experiments  on  the  same  turbine,  with  and  without  a  dif- 
fuser,  show  a  gain  in  the  coefficient  of  effect,  due  to  the  latter, 
of  about  three  per  cent.  By  the  principles  of  living  forces,  and 
assuming  that  the  motion  of  the  water  is  free  from  irregularity, 
the  gain  should  be  about  five  per  cent.  The  difference  is  due, 
in  part  at  least,  to  the  unstable  equilibrium  of  water  flowing 
through  expanding  apertures  ;  this  must  interfere  with  the  uni- 
formity of  the  velocities  of  the  fluid  streams,  at  equal  distances 
from  the  wheel. 

Suspending  the  wheel  on  the  top  of  the  vertical  shaft  (Fig. 
A),  instead  of  running  it  on  a  step  at  the  bottom.  This  had  been 
previously  attempted,  but  not  with  such  success  as  to  warrant 
its  general  adoption.  It  has  been  accomplished  with  complete 
success  by  Mr.  Boyden,  whose  mode  is  to  cut  the  upper  part  of 
the  shaft  into  a  series  of  necks,  and  to  rest  the  projecting  parts 
upon  corresponding  parts  of  a  box.  A  proper  fit  is  secured  by 
lining  the  box,  which  is  of  cast-iron,  with  Babbitt  metal — a  soft 
metallic  composition  consisting,  principally,  of  tin;  the  cast- 
iron  box  is  made  with  suitable  projections  and  recesses,  to  sup- 
port and  retain  the  soft  metal,  which  is  melted  and  poured  into 
it,  the  shaft  being  at  the  same  time  in  its  proper  position  in  the 
box.  It  will  readily  be  seen  that  a  great  amount  of  bearing- 
surface  can  be  easily  obtained  by  this  mode,  and  also,  what  is 
of  equal  importance,  it  may  be  near  the  axis ;  the  lining  metal, 
being  soft,  yields  a  little  if  any  part  of  the  bearing  should  re- 
ceive a  great  excess  of  weight.  The  cast-iron  box  is  suspended 
on  gimbals,  similar  to  those  usually  adopted  for  mariners'  com- 


APPENDIX.  147 

passes  and  chronometers,  which  arrangement  permits  the  box 
to  oscillate  freely  in  all  directions,  horizontally,  and  prevents, 
in  a  great  measure,  all  danger  of  breaking  the  shaft  at  the 
necks,  in  consequence  of  imperfections  in  the  workmanship  or 
in  the  adjustments.  Several  years'  experience  has  shown  that 
this  arrangement,  carefully  constructed,  is  all  that  can  be  de- 
sired ;  and  that  a  bearing  thus  constructed  is  as  durable,  and 
can  be  as  readily  oiled  and  taken  care  of,  as  any  of  the  ordinary 
bearings  in  a  manufactory. 

The  buckets  are  secured  to  the  crowns  of  the  wheel  in  a 
novel  and  much  more  perfect  manner  than  had  been  previously 
used  ;  the  crowns  are  first  turned  to  the  required  form,  and 
made  smooth  ;  by  ingenious  machinery  designed  for  the  pur- 
pose, grooves  are  cut  with  great  accuracy  in  the  crowns,  of  the 
exact  curvature  of  the  buckets ;  mortices  are  cut  through  the 
crowns  in  several  places  in  each  groove ;  the  buckets,  or  floats, 
are  made  with  corresponding  tenons,  which  project  through  the 
crowns,  and  are  riveted  on  the  bottom  of  the  lower  crown,  and 
on  the  top  of  the  upper  crown  ;  this  construction  gives  the  re- 
quisite strength  and  firmness,  with  buckets  of  much  thinner 
iron  than  was  necessary  under  any  of  the  old  arrangements ;  it 
also  leaves  the  passages  through  the  wheel  entirely  free  from 
injurious  obstructions. 

In  the  year  1849,  the  manufacturing  companies  at  Lowell 
purchased  of  Mr.  Boyden  the  right  to  use  all  his  improvements 
relating  to  turbines  and  other  hydraulic  motors.  Since  that 
time  it  has  devolved  upon  the  author,  as  the  chief  engineer  of 
these  companies,  to  design  and  superintend  the  construction  of 
such  turbines  as  might  be  wanted  for  their  manufactories,  and 
to  aid  him  in  this  important  undertaking,  Mr.  Boyden  has 
communicated  to  him  copies  of  many  of  his  designs  for 

turbines,  together  with  the  results  of  experiments  upon  a  por- 
10 


148  APPENDIX. 

tion  of  them ;  he  lias  communicated,  however,  but  little  theo- 
retical information,  and  the  author  has  been  guided  principally 
by  a  comparison  of  the  most  successful  designs,  and  such  light 
as  he  could  obtain  from  writers  on  this  most  intricate  subject. 

Summary  description  of  one  of  the  turbines  at  the  Tremont 
Mills,  Lowell.  Figs.  A,  B,  C. 

Fig.  A  is  a  vertical  section  of  the  turbine  through  the  axis  of  the  wheel 
shaft ;  Fig.  B  is  a  portion  of  the  plan,  on  an  enlarged  scale,  showing  the 
disposition  of  the  leading  curves  and  buckets  and  diffuser ;  Fig.  C  is  a  cross 
section  of  the  wheel  and  diffuser  on  an  enlarged  scale,  and  the  more  adja- 
cent parts.  The  letters  on  the  corresponding  parts  of  the  figures  are  the 
same. 

The  water  is  conveyed  to  the  wheel  of  the  turbine,  from  the 
forebay  by  a  supply  pipe,  the  greater  portion  of  which,  from 
the  forebay  downwards,  is  of  wrought  iron,  and  of  gradually 
diminishing  diameter  towards  the  lower  portion  I,  termed  the 
curbs,  which  is  of  cast  iron.  The  curbs  are  supported  on  col- 
umns, which  rest  on  cast-iron  supports  firmly  imbedded  in  the 
wheel-pit. 

The  Disc  K,  K',  K",  to  which  the  guides  for  the  water,  or 
the  leading  curves,  thirty -three  in  number,  are  attached,  is  sus- 
pended from  the  upper  end  of  the  cast-iron  curb,  by  means  of 
the  disc-pipes  M  M. 

The  leading  curves  are  of  Russian  iron,  one-tenth  of  an  inch 
in  thickness.  The  upper  corners  of  these,  near  the  wheel,  are 
connected  by  what  is  termed  the  garniture  L,  I/,  L",  intended 
to  diminish  the  contraction  of  the  fluid  vein  when  the  regulat- 
ing gate  is  fully  raised. 

The  disc-pipe  is  very  securely  fastened,  to  sustain  the  pressure 
of  the  water  on  the  disc.  The  escape  of  water,  between  the 
upper  curb  and  the  upper  flange  of  the  disc-pipe,  is  prevented 
by  a  band  of  leather  on  the  outside,  enclosed  within  an  iron 


APPENDIX.  14-9 

ring.  This  pipe  is  so  fastened  as  to  prevent  its  rotating  in  a 
direction  opposite  to  that  in  which  the  water  flows  out. 

The  regulating  gate  is  a  cast-iron  cylinder,  R,  enclosing  the 
disc  and  curves,  and  which,  raised  or  lowered  by  suitable 
machinery,  regulates  the  amount  of  water  let  on  the  wheel  B 
B'  B",  exterior  to  it. 

The  wheel  consists  of  a  central  plate  of  cast-iron  and  two 
crowns,  C,C,  C',  C",  of  the  same  material  to  which  the  buckets 
are  attached.  These  pieces  are  all  accurately  turned,  and  pol- 
ished, to  offer  the  least  obstruction  in  revolving  rapidly  in  the 
water. 

The  buckets,  made  of  Russian  iron,  are  forty-four  in  number, 
and  each  -fa  of  an  inch  thick.  They  are  firmly  fastened  to  the 
crowns. 

The  vertical  shaft  D,  from  which  motion  is  communicated 
to  the  machinery  by  suitable  gearing,  is  of  wrought-iron.  In- 
stead of  resting  on  a  gudgeon,  or  step  at  bottom,  it  is  suspended 
from  a  suspension  box,  E',  by  which  the  collars  at  the  top  are  en- 
closed. These  collars  are  of  steel,  and  are  fastened  to  the  upper 
portion  of  the  shaft,  which  last  can  be  detached  from  the  lower 
portion. 

The  suspension  ~box  is  lined  with  Babbitt  metal,  a  soft  compo- 
sition consisting  mostly  of  tin,  and  capable  of  sustaining  a 
pressure  of  from  50  Ibs.  to  100  Ibs.  per  square  inch,  without 
sensible  diminution  of  durability.  The  box  consists  of  two 
parts,  for  the  convenience  of  fastening  it  on,  or  the  reverse. 
The  box  rests  upon  the  gimbal  G,  which  is  so  arranged  that  the 
suspension  box,  the  shaft,  and  the  wheel  can  be  lowered  or 
raised,  and  the  suspension  box  be  allowed  to  oscillate  laterally, 
so  as  to  avoid  subjecting  it  to  any  lateral  strain. 

The  lower  end  of  the  shaft  has  a  cast-steel  pin,  O,  fixed  to  it. 
This  is  retained  in  its  place  by  the  step,  which  is  made  of  three 


150  APPENDIX. 

parts,  and  lined  with  case-hardened  iron.  The  step  can  be  ad- 
justed by  horizontal  screws,  by  a  small  lateral  motion  given  by 
them  to  it. 

Rules  for  proportioning  turbines.  In  making  the  designs 
for  the  Tremont  and  other  turbines,  the  author  has  been  guided 
by  the  following  rules,  which  he  has  been  led  to,  by  a  compari- 
son of  several  turbines  designed  by  Mr.  Boy  den,  which  have 
been  carefully  tested,  and  found  to  operate  well. 

Rule  1st.  The  sum  of  the  shortest  distances  between  the 
buckets  should  be  equal  to  the  diameter  of  the  wheel. 

Rule  2d.  The  height  of  thfe  orifices  of  the  circumference  of 
the  wheel  should  be  equal  to  one-tenth  of  the  diameter  of  the 
wheel. 

Rule  3d.  The  width  of  the  crowns  should  be  four  times  the 
shortest  distance  between  the  buckets. 

Rule  4th.  The  sum  of  the  shortest  distances  between  the 
curved  guides,  taken  near  the  wheel,  should  be  equal  to  the  in- 
terior diameter  of  the  wheel. 

The  turbines,  from  a  comparison  of  which  the  above  rules 
were  derived,  varied  in  diameter  from  twenty-eight  inches  to 
nearly  one  hundred  inches,  and  operated  on  falls  from  thirty 
feet  to  thirteen  feet.  The  author  believes  that  they  may  be 
safely  followed  for  all  falls  between  five  feet  and  forty  feet,  and 
for  all  diameters  not  less  than  two  feet ;  and,  with  judicious 
arrangements  in  other  respects,  and  careful  workmanship,  a 
useful  effect  of  seventy-five  per  cent,  of  the  power  expended 
may  be  relied  upon.  For  falls  greater  than  forty  feet,  the  sec- 
ond rule  should  be  modified,  by  making  the  height  of  the 
orifices  smaller  in  proportion  to  the  diameter  of  the  wheel. 

Taking  the  foregoing  rules  as  a  basis,  we  may,  by  aid  of  the 
experiments  on  the  Tremont  turbine,  establish  the  following 
formulas.  Let 


APPENDIX.  151 

D  =  the  diameter  of  the  wheel  at  the  outer  extremities  of 
the  buckets. 

d  =  the  diameter  at  their  inner  extremities. 

H  —  the  height  of  the  orifices  of  discharge,  at  the  outer  ex- 
tremities of  the  buckets. 

W  =  width  of  crowns  of  the  buckets. 

N  =  the  number  of  buckets. 

n  —  the  number  of  guides. 

P  =  the  horse-power  of  the  turbine,  of  550  lbfi-  *• 

h  —  the  fall  acting  on  the  wheel. 

Q  —  the  quantity  of  water  expended  by  the  turbine,  in  cubic 
feet  per  second. 

Y=  the  velocity  due  the  fall  acting  on  the  wheel. 

V  =  the  velocity  of  the  water  passing  the  narrowest  sections 
of  the  wheel. 

v  =  the  velocity  of  the  interior  circumference  of  the  wheel ; 
all  velocities  being  in  feet  per  second. 

G=  the  coefficient  of  V' ',  or  the  ratio  of  the  real  velocity 
of  the  water  passing  the  narrowest  sections  of  the  wheel,  to  the 
theoretical  velocity  due  the  fall  acting  on  the  wheel. 

The  unit  of  length  is  the  English  foot. 

It  is  assumed  that  the  useful  effect  is  seventy-five  per  cent, 
of  the  total  power  of  the  water  expended. 

According  to  Rule  1st,  we  have  the  sum  of  the  widths  of  the 
orifices  of  discharge,  equal  to  D.  Then  the  sum  of  the  areas 
of  all  the  orifices  of  discharge  is  equal  to  D  H. 

By  the  fundamental  law  of  hydraulics,  we  have 

V  =  ^ 
Therefore  V  =  C 

For  the  quantity  of  water  expended  we  have 
Q  =  HDV  =  HD C  V^JH. 

From  the  extremely  interesting  and  accurate  experiments  of 


152  APPENDIX. 

Mr.  Francis  on  the  expenditure  of  water  by  one  of  the  Tremont 
wheels,  recorded  in  his  work,  the  following  data  are  obtained 
from  it  :  — 

For  the  sum  of  the  widths  of  the  orifices  of  discharge, 

44  x  0.18757  =  8.25308  feet. 
Q  =  138.1892  cubic  feet  per  second; 
h  =  12.903  feet; 
V2g  =  8.0202  feet. 

Substituting  these  numerical  results  in  the  preceding  value 
of  Q,  there  obtains 

138.1892  =  7.68692  x  8.0202  1/12.903  <7, 
hence 

0  =  0.624. 
By  Rule  2d  we  have 

H=  0.10  Z>,  hence  HD  =  0.10  D\ 
hence  Q  =  H  D  V  =  0.10  Z>2  O  V~2g~L 

Calling  the  weight  of  a  cubic  foot  of  water  62.33  Ibs.,  we 
have 


550 
or,  substituting  for  Q  the  value  just  found, 

P=  0.0425  D*  h 
hence 


Af 

The  number  of  buckets  is  to  a  certain  extent  arbitrary,  and 
would  usually  be  determined  by  practical  considerations.  Some 
of  the  ideas  to  be  kept  in  mind  are  the  following  : 

The  pressure  on  each  bucket  is  less,  as  the  number  is  greater  ; 
the  greater  number  will  therefore  permit  of  the  use  of  thinner 
iron,  which  is  important  in  order  to  obtain  the  best  results. 
The  width  of  the  crowns  will  be  less  for  a  greater  number  of 


APPENDIX.  153 

buckets.  A  narrow  crown  appears  to  be  favorable  to  the  useful 
effect,  when  the  gate  is  only  partially  raised.  As  the  spaces 
between  the  buckets  must  be  proportionally  narrower  for  a 
larger  number  of  buckets,  the  liability  to  become  choked  up, 
either  with  anchor  ice  or  other  substances,  is  increased.  The 
amount  of  power  lost  by  the  friction  of  the  water  against  the 
surfaces  of  the  buckets  will  not  be  materially  changed,  as  the 
total  amount  of  rubbing  surface  on  the  buckets  will  be  nearly 
constant  for  the  same  diameter ;  there  will  be  a  little  less  on 
the  crown,  for  the  larger  number.  The  cost  of  the  wheel  will 
probably  increase  with  the  number  of  buckets.  The  thickness 
and  quality  of  the  iron,  or  other  metal  intended  to  be  used  for 
the  buckets,  will  sometimes  be  an  element.  In  some  water 
wrought  iron  is  rapidly  corroded. 

The  author  is  of  opinion  that  a  general  rule  cannot  be  given 
for  the  number  of  buckets ;  among  the  numerous  turbines  work- 
ing rapidly  in  Lowell,  there  are  examples  in  which  the  shortest 
distance  between  the  buckets  is  as  small  as  0.75  of  an  inch,  and 
in  others  as  large  as  2.75  inches. 

As  a  guide  in  practice,  to  be  controlled  by  particular  circum- 
stances, the  following  is  proposed,  to  be  limited  to  diameters 
of  not  less  than  two  feet : — 

^=300 +  10). 
Taking  the  nearest  whole  number  for  the  value  of  N. 

The  Tremont  turbine  is  8£  in  diameter,  and,  according  to  the 
proposed  rule,  should  have  fifty-five  buckets  instead  of  forty- 
four.  With  fifty-five  buckets,  the  crowns  should  have  a  width 
of  7.2  inches  instead  of  9  inches.  With  the  narrower  width,  it 
is  probable  that  the  useful  effect,  in  proportion  to  the  power 
expended,  would  have  been  a  little  greater  when  the  gate  was 
partially  raised. 

By  the  3d  rule,  we  have  for  the  width  of  the  crowns, 


154  APPENDIX. 


and  for  the  interior  diameter  of  the  wheel, 
,  SD 


By  the  4th  rule,  d  is  also  equal  to  the  sum  of  the  shortest 
distances  between  the  guides,  where  the  water  leaves  them. 

The  number  n  of  the  guides  is,  to  a  certain  extent,  arbitrary. 
The  practice  at  Lowell  has  been,  usually,  to  have  from  a  half  to 
three-fourths  of  the  number  of  the  buckets  ;  exactly  half  would 
probably  be  objectionable,  as  it  would  tend  to  produce  pulsa- 
tions or  vibrations. 

The  proper  velocity  to  be  given  to  the  wheel  is  an  impor- 
tant consideration.  Experiment  30  (the  one  above  used  for 
data)  on  the  Tremont  turbine  gives  the  maximum  coefficient 
of  effect  of  that  wheel  ;  in  that  experiment,  the  velocity  of  the 
interior  circumference  of  the  wheel  is  0.62645  of  the  velocity 
due  to  the  fall  acting  on  the  wheel.  By  reference  to  other  ex- 
periments, with  the  gate  fully  raised,  it  will  be  seen,  however, 
that  the  coefficient  of  effect  varies  only  about  two  per  cent. 
from  the  maximum,  for  any  velocity  of  the  interior  circumfer- 
ence, between  fifty  per  cent,  and  seventy  per  cent,  of  chat  due 
to  the  fall  acting  upon  the  wheel.  By  reference  to  the  experi- 
ments in  which  the  gate  is  only  partially  raised,  it  will  be  seen 
that  the  maximum  corresponds  to  slower  velocities  ;  and  as  tur- 
bines, to  admit  being  regulated  in  velocity  for  variable  work, 
must,  almost  necessarily,  be  used  with  a  gate  not  fully  raised, 
it  would  appear  proper  to  give  them  a  velocity  such  that  they 
will  give  a  good  effect  under  these  circumstances. 

With  this  view,  the  following  is  extracted  from  the  experi- 
ments in  Table  II.  :  — 


APPENDIX. 


155 


Number  of  the  ex- 
periment. 

Height  of  the  regulat- 
ing gate  in  inches. 

Eatio  of  the  velocity  of  the  in- 
terior circumference  of  the 
wheel,  to  the  velocity  due  the 
fall  acting  upon  the  wheel,  cor- 
responding to  the  maximum 
coefficient  of  effect. 

30 

11.49 

0.62645 

62 

8.55 

0.56541 

73 

5.65 

0.56205 

84 

2.875 

0.48390 

By  this  table  it  would  appear  that,  as  turbines  are  generally 
used,  a  velocity  of  the  interior  circumference  of  the  wheel,  of 
about  fifty-six  per  cent,  of  that  due  to  the  fall  acting  upon  the 
wheel,  would  be  most  suitable.  By  reference  to  the  diagram 
at  Plate  YI,*  it  will  be  seen  that  at  this  velocity,  when  the 
gate  is  fully  raised,  the  coefficient  of  effect  will  be  within  less 
than  one  per  cent,  of  the  maximum. 

Other  considerations,  however,  must  usually  be  taken  into  ac- 
count in  determining  the  velocity  ;  the  most  frequent  is  the  vari- 
ation of  the  fall  under  which  the  wheel  is  intended  to  operate. 
If,  for  instance,  it  were  required  to  establish  a  turbine  of  a  given 
power  on  a  fall  liable  to  be  diminished  to  one-half  by  back- 
water, and  that  the  turbine  should  be  of  a  capacity  to  give  the 
requisite  power  at  all  times,  in  this  case  the  dimensions  of  the 
turbine  must  be  determined  for  the  smallest  fall ;  but  if  it  has 
assigned  to  it  a  velocity,  to  give  the  maximum  eifect  at  the 
smallest  fall,  it  will  evidently  move  too  slow  for  the  greatest 
fall,  and  this  is  the  more  objectionable,  as,  usually,  when  the 
fall  is  greatest  the  quantity  of  water  is  the  least,  and  it  is  of 
the  most  importance  to  obtain  a  good  eifect.  It  would  then  be 
*  "  Lowell  Hydraulic  Experiments." 


156  APPENDIX. 

usually  the  best  arrangement  to  give  the  wheel  a  velocity  cor- 
responding to  the  maximum  coefficient  of  effect,  when  the  fall 
is  greatest.  To  assign  this  velocity,  we  must  find  the  propor- 
tional height  of  the  gate  when  the  fall  is  greatest  ;  this  may  be 
determined  approximately  by  aid  of  the  experiments  on  the 
Tremont  turbine. 

We  have  seen  that  P=  0.085  Qh. 

Now,  if  h  is  increased  to  2  A,  the  velocity,  and  consequently 
the  quantity,  of  water  discharged  will  be  increased  in  the  pro- 
portion of  V  h  to  \/2A  ;  that  is  to  say,  the  quantity  for  the  fall- 
2A  will  be  V%  Q. 

Calling  P'  the  total  power  of  the  turbine  on  the  double  fall, 
we  have 


or, 

P'=0.085  x  2.8284  #  h. 

Thus,  the  total  power  of  the  turbine  is  increased  2.8284  times, 
by  doubling  the  fall  ;  on  the  double  fall,  therefore,  in  order  to 
preserve  the  effective  power  uniform,  the  regulating  gate  must 
be  shut  down  to  a  point  that  will  give  only  y.-g-J-g-y  part  of  the 
total  power  of  the  turbine. 

In  Experiment  15,  the  fall  acting  upon  the  wheel  was  12.888 
feet,  and  the  total  useful  effect  of  the  turbine  was  85625.3 
Ibs.  raised  one  foot  per  second  ;  ^-.-gVsr  Par*  °f  this  is  30273.4 
Ibs.  ;  consequently  the  same  opening  of  gate  that  would  give 
this  last  power  on  a  fall  of  12.888  feet,  would  give  a  power  of 
85625.3  Ibs.  raised  one  foot  per  second,  on  a  fall  of  2x12.888 
feet=25.776  feet.  To  find  this  opening  of  gate,  we  must  have 
recourse  to  some  of  the  other  experiments. 

In  Experiment  73,  the  fall  was  13.310  feet,  the  height  of  the 
gate  5.65  inches,  and  the  useful  effect  58830.1  Ibs.  In  Ex- 
periment 83  the  fall  was  13.435  feet,  the  height  of  the  gate 


APPENDIX.  157 

2.875  inches,   and  the  useful  effect  2T310.9  Ibs.      Reducing 

o 

both  these  useful  effects  to  what  they  would  have  been  if  the 
fall  was  12.888  feet, 

the  useful  effect  in  experiment  73,  58830.l(— -— ^=56054.5; 

vlo.olO/ 

"       "         "       "          "  83,  27310.9(12-888)£=25660.1. 

>J.O.4:OO' 

By  a  comparison  of  the  useful  effects  with  the  corresponding 
heights  of  gate,  we  find,  by  simple  proportion  of  the  differences, 
that  a  useful  effect  of  30273.4  Ibs.  raised  one  foot  high  per 
second,  would  be  given  when  the  height  of  the  regulating  gate 
was  3.296  inches. 

By  another  mode  : — 

As  25660.1 :  2.875 : :  30273.4  :  2.875  x  ffflf  :f =3.392  in.,  a  little 
consideration  will  show  that  the  first  mode  must  give  too  little, 
and  the  second  too  much  ;  taking  a  mean  of  the  two  results,  we 
have  for  the  height  of  the  gate,  giving  7<  8*2  a  4  of  the  total  power 
of  the  turbine,  3.344  inches.  Eeferring  to  Table  IL,  we  see 
that,  with  this  height  of  gate,  in  order  to  obtain  the  best  coeffi- 
cient of  useful  effect,  the  velocity  of  the  interior  circumference 
of  the  wheel  should  be  about  one-half  of  that  due  to  the  fall 
acting  upon  the  wheel ;  and  by  comparison  of  Experiments  74 
and  84,  it  will  be  seen  that,  with  this  height  of  gate  and  with 
this  velocity,  the  coefficient  of  useful  effect  must  be  near  0.50. 

This  example  shows,  in  a  strong  light,  the  well-known  defect 
of  the  turbine,  viz.,  giving  a  diminished  coefficient  of  useful 
effect  at  times  when  it  is  important  to  obtain  the  best  results. 
One  remedy  for  this  defect  would  be,  to  have  a  spare  turbine, 
to  be  used  when  the  fall  is  greatly  diminished ;  this  arrangement 
would  permit  the  principal  turbine  to  be  made  nearly  of  the  di- 
mensions required  for  the  greatest  fall.  As  at  other  heights  of 
the  water  economy  of  water  is  usually  of  less  importance, 


158  APPENDIX. 

the  spare  turbine  might  generally  be  of  a  cheaper  construc- 
tion. 

To  lay  out  the  curve  of  the  buckets,  the  author  makes  use  of 
the  following  method  :— 

Referring  to  Fig.  D,  the  number  of  buckets,  N,  having  been 

determined  by  the  preceding  rules,  set  off  the  arc  GL=^— . 

Let  u  =  GH  =  I  P',  the  shortest  distance  between  the  buckets ; 

t  —  the  thickness  of  the  metal  forming  the  buckets. 

Make  the  arc  G  K  =  5  w.  Draw  the  radius  O  K,  intersecting 
the  interior  circumference  of  the  wheel  at  L ;  the  point  L  will 
be  the  inner  extremity  of  the  bucket.  Draw  the  directrix  L  M 
tangent  to  the  inner  circumference  of  the  wheel.  Draw  the 
arc  O  N,  with  the  radius  w  +  tf,  from  I  as  a  centre ;  the  other 
directrix,  G  P,  must  be  found  by  trial,  the  required  conditions 
being,  that,  when  the  line  M  L  is  revolved  round  to  the  position 
G  T,  the  point  M  being  constantly  on  the  directrix  G  P,  and 
another  point  at  the  distance  M  G  =  R  S,  from  the  extremity 
of  the  line  describing  the  bucket,  being  constantly  on  the  di- 
rectrix M  L,  the  curve  described  shall  just  touch  the  arc  N  O. 
A  convenient  line  for  a  first  approximation  may  be  drawn  by 
making  the  angle  O  G  P  —  11°.  After  determining  the  direc- 
trix according  to  the  preceding  method,  if  the  angle  O  G  P 
should  be  greater  than  12°,  or  less  than  10°,  the  length  of  the 
arc  GK  should  be  changed  to  bring  the  angle  within  these 
limits. 

The  curve  G  S  S7  S"  L,  described  as  above,  is  nearly  the  quar- 
ter of  an  ellipse,  and  would  be  precisely  so  if  the  angle  G  M  L 
was  a  right  angle ;  the  curve  may  be  readily  described,  me- 
chanically, with  an  apparatus  similar  to  the  elliptic  trammel; 
there  is,  however,  no  difficulty  in  drawing  it  by  a  series  of 
points,  as  is  sufficiently  obvious. 


APPENDIX.  159 

The  trace  adopted  by  the  author  for  the  corresponding  guides 
is  as  follows  :  — 

The  number  n  having  been  determined,  divide  the  circle  in 
which  the  extremities  of  the  guides  are  found  into  n  equal 
parts  Y  W,  W  X,  etc. 

Put  w'  for  the  width  between  two  adjoining  guides, 

and  if  for  the  thickness  of  the  metal  forming  the  guides. 

d 
We  have  by  Rule  4,  w'  =  — 

J  n 

With  W  as  a  centre,  and  the  radius  w'  +  £',  draw  the  arc 
Y  Z  ;  and  with  X  as  a  centre,  and  the  radius  2(w'  +  tf'),  draw 
the  arc  A'  B'.  Through  Y  draw  the  portion  of  a  circle,  Y  C', 
touching  the  arcs  Y  Z  and  A'  B'  ;  this  will  be  the  curve  for  the 
essential  portion  of  the  guide.  The  remainder  of  the  guide, 
C'  D',  should  be  drawn  tangent  to  the  curve  C'  Y;  a  convenient 
radius  is  one  that  would  cause  the  curve  C'  Dx,  if  continued  to 
pass  through  the  centre  O.  This  part  of  the  guide  might  be 
dispensed  with,  except  that  it  affords  great  support  to  the  part 
C'  Y,  and  thus  permits  the  use  of  much  thinner  iron  than  would 
be  necessary  if  the  guide  terminated  at  C',  or  near  it. 

Collecting  together  the  foregoing  formulas  for  proportioning 
turbines,  which,  it  is  understood,  are  to  be  limited  to  falls  not 
exceeding  forty  feet,  and  to  diameters  not  less  than  two  feet, 
we  have  for  the  horse  power, 

P  =  OMMZPh,  tfK  ; 

for  the  diameter, 


for  the  quantity  of  water  discharged  per  second, 


160  APPENDIX. 

for  the  velocity  of  the  interior  circumference  of  the  wheel,  when 
the  fall  is  not  very  variable, 

v  =  0.56 


or,  ^  =  4.491  1/  h; 

for  the  height  of  the  orifices  of  discharge, 

H  =0.10  D; 
for  the  number  of  buckets, 

jr=3(Z>  +  10); 
for  the  shortest  distance  between  two  adjacent  buckets, 

D 
"=N> 

for  the  width  of  the  crown  occupied  by  the  buckets, 


for  the  interior  diameter  of  the  wheel, 

//      n     8J) 

--D--N*> 

for  the  number  of  guides, 

w  =  0.50  -ZT  to  0.75  N  ; 
for  the  shortest  distance  between  two  adjacent  guides, 

</=£ 

n  ' 

Table  has  been  computed  by  these  formulas. 
For  falls  greater  than  forty  feet,  the  height  of  the  orifices  in 
the  circumference  of  the  wheel  should  be  diminished.  The 
foregoing  formulas  may,  however,  still  be  made  use  of.  Thus  : 
supposing,  for  a  high  fall,  it  is  determined  to  make  the  orifices 
three-fourths  of  that  given  by  the  formula  ;  divide  the  given 
power,  or  quantity  of  water  to  be  used,  by  0.75,  and  use  the 
quotient  in  place  of  the  true  power  or  quantity,  in  determining 
the  dimensions  of  the  turbine.  No  modifications  of  the  dimen- 
sions will  be  necessary,  except  that  -jV  of  the  diameter  of  the 


APPENDIX.  161 

t 

turbine  should  be  diminished  to  •£$  of  the  diameter,  to  give  the 
height  of  the  orifices  in  the  circumference. 

It  is  plain,  from  the  method  by  which  the  preceding  formulas 
have  been  obtained,  that  they  cannot  be  considered  as  estab- 
lished, but  should  only  be  taken  as  guides  in  practical  applica- 
tions, until  some  more  satisfactory  are  proposed,  or  the  intrica- 
cies of  the  turbine  have  been  more  fully  unravelled.  The 
turbine  has  been  an  object  of  deep  interest  to  many  learned 
mathematicians,  but  up  to  this  time  the  results  of  their  investi- 
gations, so  far  as  they  have  been  published,  have  afforded  but 
little  aid  to  hydraulic  engineers. 

Diffuser. — As  previously  stated,  the  principles  involved  in 
the  flow  of  water  through  a  diverging  tube  find  a  useful  appli- 
cation in  Mr.  Boyden's  diffuser.  This  invention,  applied  to  a 
turbine  water  wheel  104.25  inches  in  diameter,  and  about  seven 
hundred  horse-power,  is  represented  on  Fig.  B  at  X'  and  on 
Fig.CatX". 

The  diffuser  is  supported  on  iron  pillars  from  below.  The 
wheel  is  placed  sufficiently  low  to  permit  the  diffuser  to  be 
submerged  at  all  times  when  the  wheel  is  in  operation,  that 
being  essential  to  the  most  advantageous  operation  of  the 
diffuser. 

When  the  speed  gate  is  fully  raised,  the  wheel  moves  with 
the  velocity  which  gives  its  greatest  coefficient  of  useful  effect. 
On  leaving  the  wheel  it  necessarily  has  considerable  velocity, 
which  would  involve  a  corresponding  loss  of  power,  except  for 
the  effect  of  the  diffuser,  which  utilizes  a  portion  of  it.  When 
operating  under  a  fall  of  thirty-three  feet,  and  the  speed  gate 
is  raised  to  its  full  height,  this  wheel  discharges  about  219 
cubic  feet  per  second.  The  area  of  the  annular  space  where 
the  water  enters  the  diffuser,  is  0.802  x  8.792  if  =  22.152  square 
feet ;  and  if  the  stream  passes  through  this  section  radially,  its 


162  APPENDIX. 

219 

mean  velocity  must  be  =  9.886  feet  per  second,  which 

22.152 

is  due  to  a  head  of  1.519  feet.     The  area  of  the  annular  space 

where  the  water  leaves  the  diff'user  is  1.5  x  15.333  if  =  72.255 

219 
square  feet,   and   the  mean   velocity  —  =3.031   feet  per 


7.2. 
second,  which  is  due  to  a  head  of  0.143  feet.    According  to  this 

the  saving  of  head  due  to  the  diffnser  is  1.519  —  0.143=1.376 

"1    Q^^i 

feet,  being  -—  --  or  about  4-|  per  cent,  of  the  head  available 

oo  —  1.519 

without  the  diffuser,  which  is  equivalent  to  a  gain  in  the  coeffi- 
cient of  useful  eifect  to  the  same  extent.  Experiments  on  the 
same  turbine,  with  and  without  the  diffuser,  have  shown  a 
gain  due  to  the  latter  of  about  three  per  cent,  in  the  coefficient 
of  useful  effect.  The  diffuser  adds  to  the  coefficient  of  useful 
effect  by  increasing  the  velocity  of  the  water  passing  through 
the  wheel,  and  it  must  of  course  increase  the  quantity  of  water 
discharged  in  the  same  proportion.  If  it  increases  the  availa- 
ble head  three  per  cent.,  the  velocity,  which  varies  as  the 
square  root  of  the  head,  must  be  increased  in  the  same  propor- 
tion. The  power  of  the  wheel,  which  varies  as  the  product  of 
the  head  into  the  quantity  of  water  discharged,  must  be 
increased  about  4.5  per  cent. 


APPENDIX.  163 


EXPLANATION  OF  FIGURES. 

Fig.  A.  Section  through  the  axis  of  the  Turbine  without  the  Diffuser. 

I  Cast-iron  Curbs  through  which  the  water  passes  from  the  wrought-iron 
supply-pipe  to  the  Disc. 

K  Cast-iron  Disc  on  which  the  Guide  Curves  are  fastened. 

L  Garniture  fitted  to  lower  end  of  Lower  Curb. 

M  Disc-pipe  suspending  the  Disc  from  Upper  Curb, 

N  Columns  of  cast-iron  supporting  Curbs. 

R  Regulating  Gate  of  cast-iron. 

5  Brackets  for  raising  and  lowering  the  Gate. 

B  Wheel. 

C,  C  Crowns  of  the  Wheel  between  which  the  curved  buckets  are  in- 
serted. 

D  Main  Shaft  of  the  Wheel. 

£'  Suspension  Box,  lined  with  babbit  metal,  from  which  the  Wheel  hangs 
by  the  cast-steel  Collars  around  the  upper  end  of  the  Shaft. 

p  Upper  portion  of  Shaft  fastened  to  lower  portion,  with  bearings  p'  of 
cast-iron  lined  with  babbit  metal. 

Q  Gimbal  on  which  the  Suspension  Box  £'  rests. 

H  Support  of  the  Gimbal. 

O  Step  to  receive  cast-steel  Pin  on  lower  end  of  Shaft. 

Fig.  A.  Plan  of  Disc,  K",  Garniture,  L',  Wheel,  C',  and  Diffuser,  M'  N'- 

Fig.  B.  Section  of  Wheel,  C",Garniture,  L">  Regulating  Gate,  R",  and 
upper  and  lower  Crowns  of  the  Diffuser. 

Fig.  D.  Diagram  for  laying  out  the  curves  of  the  Buckets  and  Guide 
Curves.  f 


164: 


APPENDIX. 


APPENDIX. 


165 


YC 


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